Optical transmitter and transmission method

ABSTRACT

An optical transmitter transmits a data signal. The optical transmitter has an encoder configured to encode the data signal by selecting based on a bit sequence a first symbol and a second symbol from a set of four symbols for each one of at least two transmission time slots. The optical transmitter further has a modulator configured to use in each transmission time slot the first symbol to modulate a first carrier wave and the second symbol to modulate a second carrier wave, and to transmit the two carrier waves over orthogonal polarizations of an optical carrier. Symbols in consecutive transmission time slots have non-identical polarization states.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a continuation of U.S. patent application Ser. No.17/402,268, filed on Aug. 13, 2021, which is a continuation of U.S.patent application Ser. No. 16/810,620, filed on Mar. 5, 2020, now U.S.Pat. No. 11,095,371, which is a continuation of InternationalApplication No. PCT/EP2018/073635, filed on Sep. 3, 2018, which claimspriority to International Patent Application No. PCT/EP2017/072258,filed on Sep. 5, 2017. All of the afore-mentioned patent applicationsare hereby incorporated by reference in their entireties.

FIELD

The disclosure relates to an optical transmitter and to a method foroptically transmitting a data signal. The disclosure specificallyrelates to the encoding and modulating of the data signal to betransmitted, and thereby especially to the use of high dimensionalmodulation formats.

BACKGROUND

In modern optical coherent transmission, information (a data signal) isencoded in the amplitude and phase of a carrier wave emitted by a laser.To this end, information bits of the data signal are mapped to symbols,which can be represented by complex numbers. A given complex symbol isthen used in turn to modulate the in-phase (I) and quadrature (Q)components of an optical carrier during a fixed time duration (referredto as ‘symbol time’, ‘interval’ or transmission ‘time slot’). Inconventional polarization-multiplexed transmission, two carrier wavesare modulated and transmitted over two orthogonal polarizations(referred to as X and Y) of the optical carrier. The optical carrier is,for instance, an optical fiber.

It has been realized that performance gains and intermediate spectralefficiencies can be achieved by designing so-called high-dimensionalmodulation formats. In particular, one can imagine each combination ofthe above-mentioned degrees of freedom as one dimension of ahigh-dimensional space. For example, the I-component of the Xpolarization may correspond to one dimension, which can be labelled IX.Then, a four-dimensional space spanned by vectors (IX, QX, IY, QY) canbe obtained. The dimension of this space can be increased e.g. to eightby grouping pairs of time slots and adding the time slot (indexed 1 or 2for two consecutive time slots) as an additional degree of freedom. Thisscheme can be generalized to even higher dimensions by increasingfurther the number of time slots to more than 2.

Selecting constellation points (so-called set partitioning) in thehigh-dimensional space corresponds to introducing constraints betweenthe different degrees of freedom. Such constraints can be used tooptimize linear performance of an optical transmitter using themodulation format. For example, linear performance can be improved bychoosing points from a lattice that corresponds to a dense packing inhigh dimensions. Nonlinear performance can, for example, be optimized bychoosing constellation points with predefined values of the polarizationstate.

However, the design of (high-dimensional) modulation formats for opticalcoherent transmission is a great technical challenge. This is due to themany boundary conditions the modulation formats have to comply with.That is, for instance, the modulation formats should meet therequirement of a maximum Required Optical Signal to Noise Ratio (ROSNR).Therefore, good performance in the linear and nonlinear channel isrequired. The ROSNR value is fixed by the type of coherent opticaltransmission system. For long haul and ultra-long haul transmissionsystems, the major limitation is the fiber Kerr effect, which impairs awavelength and polarization multiplexed optical signal by introducingSelf Phase Modulation (SPM), Cross Phase Modulation (XPM) and CrossPolarization Modulation (XPolM). Accordingly, one technical challengehere is reducing (mitigating) these effects on the optical signal.

Furthermore, in the context of the present disclosure, the modulationformats should be able to operate at intermediate spectral efficienciesbetween 2 and 4 bits/interval, wherein the projection onto an intervalcomprises the four dimensions: in-phase (I) and quadrature (Q) each intwo orthogonal polarizations (X, Y). The modulation formats should alsobe constructed with a simple mapping, so that they can be implementedwith low complexity in an optical transmitter or transmission system.Finally, the modulation formats should have optimal labeling of theconstellation points to result in a good linear performance of theoptical transmission system.

The first two modulation formats that were used in this context arePolarization Division Multiplexed Binary Phase Shift Keying andQuadrature Phase Shift Keying (PDM-BPSK and PDM-QPSK), which operate atspectral efficiencies of 2 and 4 bits/interval, respectively. Thesemodulation formats are designed in four dimensions: I and Q in each oftwo orthogonal polarizations. These symbols have constant modulus. Thismeans that the power of a symbol (the signal that is used to drive themodulator) including both polarizations does not vary in time. Thisproperty contributes to mitigating fiber nonlinear impairments comparedto legacy systems using the On-Off Keying (OOK) modulation format with atime-varying power envelope.

Later the Polarization Switched (PS) Quadrature Phase Shift Keying(PS-QPSK) modulation format was proposed, which reaches an intermediatespectral efficiency of 3 bits/interval, and exhibits good linear andnonlinear performances. The idea underlying PS-QPSK for enhancing thenonlinear performance is that of polarization switching. Thepolarization state can only take two values for any four-dimensionalsymbol. As a result the averaged polarization-rotation and thereforeXPolM caused by one signal on another are reduced. These modulationformats also have a very simple mapping. The linear performance(performance of the modulation format used in an optical transmitterunder linear transmission conditions) depends on the labeling, that is,the assignment of bit sequences to constellation points. For thementioned modulation formats optimal performance is achieved by usingstandard Gray-labeling.

However, with technology advancing further, the PDM-BPSK, PS-QPSK andPDM-QPSK nonlinear performances are not acceptable anymore, and need tobe enhanced. The following describes how this problem was addressed inthe state of the art, and what the disadvantages of the proposedconventional solutions are.

In the conventional solutions, 8-dimensional modulation formats at 2.5,3 and 3.5 bits/interval were proposed. The conventional solutions usedthe known fact that high nonlinear tolerance is achieved by designingmodulation formats with a constant modulus. That is, the power of thesymbols in all time slots is the same. In addition to this constraint,the conventional solutions proposed using the condition of polarizationbalancing (PB) in consecutive time slots, i.e. the sum of Stokes-vectorsover all symbols in time vanishes. Such a property reduces the XPolMeffects.

The modulation formats of the conventional solutions are constructed adhoc, for example, by choosing a 4 dimensional (4D) base format withconstant modulus, replicating it in subsequent time slots, and applyingthe polarization balance criterion. This can be exemplified by theconstruction of the following three modulation formats.

In order to obtain PB-PS-QPSK symbols in a first conventional solution,a set partitioning method was used based on a set of symbols offered byPS-QPSK. In 8 dimensions (8D), the latter offers 64 symbols, whichcorresponds to a spectral efficiency of 3 bits/interval. Using apolarization balance criterion, the conventional solution set partitionsPS-QPSK to obtain 32 symbols in 8 dimensions, hence a spectralefficiency of 2.5 bits/interval.

For the PB-PM-QPSK modulation format, a second conventional solutionuses QPSK symbols for each of the two polarizations on the first timeslot, and one of the polarizations on the second time slot. Then, ituses a formula to obtain the symbol of the remaining polarization on thesecond time slot. The formula fixes the constraint of the degree ofpolarization of a symbol, which is equal to zero in two consecutive timeslots. Therefore, the obtained symbols are polarization balanced in twoconsecutive time slots. With this constraint, the maximum number ofsymbols is 64 in 8D, which represents a spectral efficiency of 3bits/interval.

Using QPSK symbols with the polarization balance property reaches atmost 3 bits/interval. In order to reach a spectral efficiency of 3.5bits/interval, a third conventional solution proposed using acombination of QPSK and 8PSK symbols, increasing the number of symbolsthat are used to obtain the modulation format. The same approach thatwas used to obtain PB-PM-QPSK was followed for this modulation format:Symbols for the two polarizations on the first time slot and onepolarization of the second time slot are chosen from QPSK and 8PSKsymbols, then the same formula as before is used to obtain the symbol ofthe remaining polarization on the second time slot. The polarizationbalance criterion is consequently satisfied and a spectral efficiency of3.5 bits/interval is reached.

The disadvantage of the above-mentioned conventional solutions is thatthe modulation format construction does not necessarily yield optimallinear or nonlinear performance, mainly because the polarization balancecriterion is too restrictive. This has two main drawbacks: Firstly, athigher spectral efficiencies, one has to use base constellations withreduced Euclidean distance to satisfy the constraint. This negativelyimpacts the linear performance of an optical transmitter when using theconventional solution. Secondly, when applying set-partitioning to aspecialized format such as PS-QPSK does not necessarily yield theoptimal nonlinear performance when used by an optical transmitter.Furthermore, simple mapping rules from bits of a data signal to symbolshave not been given.

SUMMARY

In view of the above-mentioned challenges and disadvantages, the presentdisclosure improves the conventional solutions for modulation formats tobe used in optical transmission systems. The present disclosure providesan optical transmitter and a transmitting method, which operate withbetter performance compared to the corresponding solutions known in theart. Thereby, high-dimensional modulation formats with a spectralefficiency ranging from 2 to 4 bits/interval, and better linear andnonlinear performances, are desired. Further, simple mapping rules fromdata signal bits to symbols should be given.

A first aspect of the disclosure provides an optical transmitter fortransmitting a data signal, comprising an encoder configured to encodethe data signal by selecting based on a bit sequence a first symbol anda second symbol from a set of four symbols for each one of at least twotransmission time slots, and a modulator configured to use in eachtransmission time slot the first symbol to modulate a first carrier waveand the second symbol to modulate a second carrier wave, and to transmitthe two carrier waves over orthogonal polarizations of an opticalcarrier, wherein symbols in consecutive transmission time slots havenon-identical polarization states.

The optical transmitter of the first aspect achieves a betterperformance than an optical transmitter according to the conventionalsolutions. While symbols ideally have opposite polarization inconsecutive time slots (referred to as ‘polarization balancing’, whichis strictly applied in the conventional solutions), the inventors of thepresent disclosure have found that it is most important to avoid symbolshaving identical states of polarization in consecutive time slots(referred to as ‘polarization identical’). Symbols which have differentpolarization states in subsequent time slots (referred to as‘polarization alternating’, PA) may potentially degrade the nonlinearperformance, when they are used in place of polarization balanced (PB)symbols. However using such symbols relaxes the PB constraint andthereby allows one to significantly improve the linear performance asdetailed below.

It is noted that in the present disclosure ‘polarization states’ are asdescribed by their corresponding Stokes vectors. That means,designations like ‘antiparallel’ ‘opposite’, ‘different’ or ‘orthogonal’polarization states refer to the relative orientation of the Stokesvectors of symbols in subsequent time slots. As known in the art, theStokes vector can be calculated given two symbols corresponding to thetwo polarizations. The result of the calculation depends on thedefinition of the Stokes vector (or the used basis). However, in thepresent disclosure the important point is that the Stokes vectors inconsecutive time slots are not identical, and in particular may be‘antiparallel’ or ‘orthogonal’. These properties do not depend on thedefinition of the Stokes vector.

The optical transmitter of the first aspect uses a set of four symbols,i.e. a common base constellation, which leads to an improved linearperformance compared to the conventional solutions known in the art.This is because the base constellation has a higher Euclidian distancethan other constellations, for instance, when compared to constellationsused in the conventional solutions. The use of the QPSK baseconstellation here is possible only because the polarization-balancecriterion is relaxed in comparison with its strict use in theconventional solutions.

By using a set of only four symbols, and not strictly using the‘polarization balancing’ but the ‘polarization alternating’ concept, theoptical transmitter of the first aspect can be used with differentmodulation formats, which all result in better linear and non-linearchannel performance at the same spectral efficiency. In particular, twoexemplary modulation formats are presented later, both havingeight-dimensional (8D) modulation formats at spectral efficiencies of2.5 and 3.5 bits/transmission time slot (bits/interval), respectively.

In an implementation form of the first aspect, the encoder is configuredto select the symbols from a QPSK base constellation.

Only the relaxing of the polarization-balance criterion allows using,for instance, the PDM-QPSK as the base constellation. In contrastthereto, it is not possible in general to use a QPSK base constellationwith the strict polarization balancing applied in the conventionalsolutions.

In a further implementation form of the first aspect, the modulator isconfigured to modulate an In-Phase and a Quadrature component of eachcarrier wave.

In a further implementation form of the first aspect, symbols in atleast a subset of consecutive transmission time slots have anti-parallelpolarization states.

The use of anti-parallel Stokes vectors in subsequent time slots, wherepossible, improves the performance of the optical transmitter.

In a further implementation form of the first aspect, the encoder isconfigured to generate the bit sequence based on the data signal, andthe data signal comprises less bits than the encoded bit sequence.

In a further implementation form of the first aspect, the encoder isconfigured to perform at least one arithmetic operation based on atleast two bits of the data signal to obtain at least one overhead bit,and to generate the bit sequence based on the bits of the data signaland at least one overhead bit.

According to the above implementation forms, overhead bits aregenerated. The use of overhead bits results implicitly in constraintsfor selecting the symbols, which leads to a better nonlinearperformance.

In a further implementation form of the first aspect, the opticaltransmitter is configured to transmit the data signal with a spectralefficiency of 2.5 bits per transmission time slot.

In a further implementation form of the first aspect, the symbolpolarization states in each transmission time slot take one of at leastfour distinct polarizations states.

This distinguishes from the conventional solutions and leads to betteroverall performance.

In a further implementation form of the first aspect, the data signalhas five bits b1 . . . b5, and the encoder is configured to generate thebit sequence having eight bits b1 . . . b5, b1′, b2′, b3′, wherein threeoverhead bits b1′, b2′, b3′ are generated according tob1′=b3XORb4XORb5b2′= b2XORb4XORb5b3′= b1XORb4XORb5

In a further implementation form of the first aspect, for twoconsecutive transmission time slots T1 and T2, for two orthogonalpolarizations X and Y of the optical carrier, and for a set of four QPSKsymbols denoted −1−1i, −1+1i, 1−1i and 1+1i, the encoder is configuredto select the symbols based on the data signal according to thefollowing labelling

Labelling (from left to right) 5 bits b1 . . . b5, and 3 overhead Timeslot T₁ Time slot T₂ bits b1′, b2′, b3′ X polarization Y polarization Xpolarization Y polarization 0 0 0 0 0 0 1 1 −1 − 1i −1 − 1i −1 − 1i  1 +1i 0 0 0 0 1 1 0 0 −1 − 1i −1 − 1i  1 + 1i −1 − 1i 0 0 0 1 0 1 0 0 −1 −1i −1 + 1i −1 + 1i −1 − 1i 0 0 0 1 1 0 1 1 −1 − 1i −1 + 1i  1 − 1i  1 +1i 0 0 1 0 0 1 1 1 −1 − 1i  1 − 1i −1 + 1i  1 + 1i 0 0 1 0 1 0 0 0 −1 −1i  1 − 1i  1 − 1i −1 − 1i 0 0 1 1 0 0 0 0 −1 − 1i  1 + 1i −1 − 1i −1 −1i 0 0 1 1 1 1 1 1 −1 − 1i  1 + 1i  1 + 1i  1 + 1i 0 1 0 0 0 0 0 1 −1 +1i −1 − 1i −1 − 1i −1 + 1i 0 1 0 0 1 1 1 0 −1 + 1i −1 − 1i  1 + 1i  1 −1i 0 1 0 1 0 1 1 0 −1 + 1i −1 + 1i −1 + 1i  1 − 1i 0 1 0 1 1 0 0 1 −1 +1i −1 + 1i  1 − 1i −1 + 1i 0 1 1 0 0 1 0 1 −1 + 1i  1 − 1i −1 + 1i −1 +1i 0 1 1 0 1 0 1 0 −1 + 1i  1 − 1i  1 − 1i  1 − 1i 0 1 1 1 0 0 1 0 −1 +1i  1 + 1i −1 − 1i  1 − 1i 0 1 1 1 1 1 0 1 −1 + 1i  1 + 1i  1 + 1i −1 +1i 1 0 0 0 0 0 1 0  1 − 1i −1 − 1i −1 − 1i  1 − 1i 1 0 0 0 1 1 0 1  1 −1i −1 − 1i  1 + 1i −1 + 1i 1 0 0 1 0 1 0 1  1 − 1i −1 + 1i −1 + 1i −1 +1i 1 0 0 1 1 0 1 0  1 − 1i −1 + 1i  1 − 1i  1 − 1i 1 0 1 0 0 1 1 0  1 −1i  1 − 1i −1 + 1i  1 − 1i 1 0 1 0 1 0 0 1  1 − 1i  1 − 1i  1 − 1i −1 +1i 1 0 1 1 0 0 0 1  1 − 1i  1 + 1i −1 − 1i −1 + 1i 1 0 1 1 1 1 1 0  1 −1i  1 + 1i  1 + 1i  1 − 1i 1 1 0 0 0 0 0 0  1 + 1i −1 − 1i −1 − 1i −1 −1i 1 1 0 0 1 1 1 1  1 + 1i −1 − 1i  1 + 1i  1 + 1i 1 1 0 1 0 1 1 1  1 +1i −1 + 1i −1 + 1i  1 + 1i 1 1 0 1 1 0 0 0  1 + 1i −1 + 1i  1 − 1i −1 −1i 1 1 1 0 0 1 0 0  1 + 1i  1 − 1i −1 + 1i −1 − 1i 1 1 1 0 1 0 1 1  1 +1i  1 − 1i  1 − 1i  1 + 1i 1 1 1 1 0 0 1 1  1 + 1i  1 + 1i −1 − 1i  1 +1i 1 1 1 1 1 1 0 0  1 + 1i  1 + 1i  1 + 1i −1 − 1i

The above-given tables and formulas define the 2.5 bits/transmissiontime slot modulation format according to the previous implementationdefines the modulation format up to symmetries and change of labels,which change the formulas to generate the overhead bits and lead toequivalent modulation formats with the same performance.

According to the above implementation forms, the modulation format atthe spectral efficiency of 2.5 bits/interval in 8D is realized. Themodulation format has a better nonlinear performance than theconventional solution with the same spectral efficiency. In particular,the modulation format has a better nonlinear performance (found to be0.35 dB higher in Q² factor), even though the linear performance is thesame. It is noted that the conventional solution for spectral efficiencyof 2.5 bit/interval has only two different polarization states, whilethe one of the above implementation forms of the first aspect has four.

In a further implementation form of the first aspect, the opticaltransmitter is configured to transmit the data signal with a spectralefficiency of 3.5 bits per transmission time slot.

In a further implementation form of the first aspect, symbols in atleast a subset of consecutive transmission time slots have orthogonalpolarization states.

The use of orthogonal Stokes vectors in subsequent time slots, wherepossible, improves the performance of the optical transmitter.

In a further implementation form of the first aspect, a portion of thedata signal has seven bits b1 . . . b7, and the encoder is configured togenerate the bit sequence having eight bits b1 . . . b7, b′, wherein theoverhead bit b′ is generated according to:b′=b1XORb4XORb6XOR(b1ANDb3)XOR(b1ANDb4)XOR(b1ANDb5)XOR(b1ANDb6)XOR(b2ANDb3)XOR(b2ANDb4)XOR(b2ANDb5)XOR(b2ANDb6)XOR(b3ANDb5)XOR(b3ANDb6)XOR(b4ANDb5)XOR(b4ANDb6)

In a further implementation form of the first aspect, for twoconsecutive transmission time slots T1 and T2, for two orthogonalpolarizations X and Y of the optical carrier, and for a set of four QPSKsymbols denoted −1−1i, −1+1i, 1−1i and 1+1i, the encoder is configuredto select the symbols based on the data signal according to thefollowing labelling

Labelling (from left to right) 7 bits b1 . . . b7, Time slot T₁ Timeslot T₂ and 1 overhead bit b′ X polarization Y polarization Xpolarization Y polarization 0 0 0 0 0 0 0 1 −1 − 1i −1 − 1i −1 − 1i −1 +1i 0 0 0 0 0 0 1 1 −1 − 1i −1 − 1i −1 − 1i  1 + 1i 0 0 0 0 0 1 0 0 −1 −1i −1 − 1i −1 + 1i −1 − 1i 0 0 0 0 0 1 1 0 −1 − 1i −1 − 1i −1 + 1i  1 −1i 0 0 0 0 1 0 0 1 −1 − 1i −1 − 1i  1 − 1i −1 + 1i 0 0 0 0 1 0 1 1 −1 −1i −1 − 1i  1 − 1i  1 + 1i 0 0 0 0 1 1 0 0 −1 − 1i −1 − 1i  1 + 1i −1 −1i 0 0 0 0 1 1 1 0 −1 − 1i −1 − 1i  1 + 1i  1 − 1i 0 0 0 1 0 0 0 0 −1 −1i −1 + 1i −1 − 1i −1 − 1i 0 0 0 1 0 0 1 0 −1 − 1i −1 + 1i −1 − 1i  1 −1i 0 0 0 1 0 1 0 0 −1 − 1i −1 + 1i −1 + 1i −1 − 1i 0 0 0 1 0 1 1 0 −1 −1i −1 + 1i −1 + 1i  1 − 1i 0 0 0 1 1 0 0 1 −1 − 1i −1 + 1i  1 − 1i −1 +1i 0 0 0 1 1 0 1 1 −1 − 1i −1 + 1i  1 − 1i  1 + 1i 0 0 0 1 1 1 0 1 −1 −1i −1 + 1i  1 + 1i −1 + 1i 0 0 0 1 1 1 1 1 −1 − 1i −1 + 1i  1 + 1i  1 +1i 0 0 1 0 0 0 0 1 −1 − 1i  1 − 1i −1 − 1i −1 + 1i 0 0 1 0 0 0 1 1 −1 −1i  1 − 1i −1 − 1i  1 + 1i 0 0 1 0 0 1 0 1 −1 − 1i  1 − 1i −1 + 1i −1 +1i 0 0 1 0 0 1 1 1 −1 − 1i  1 − 1i −1 + 1i  1 + 1i 0 0 1 0 1 0 0 0 −1 −1i  1 − 1i  1 − 1i −1 − 1i 0 0 1 0 1 0 1 0 −1 − 1i  1 − 1i  1 − 1i  1 −1i 0 0 1 0 1 1 0 0 −1 − 1i  1 − 1i  1 + 1i −1 − 1i 0 0 1 0 1 1 1 0 −1 −1i  1 − 1i  1 + 1i  1 − 1i 0 0 1 1 0 0 0 0 −1 − 1i  1 + 1i −1 − 1i −1 −1i 0 0 1 1 0 0 1 0 −1 − 1i  1 + 1i −1 − 1i  1 − 1i 0 0 1 1 0 1 0 1 −1 −1i  1 + 1i −1 + 1i −1 + 1i 0 0 1 1 0 1 1 1 −1 − 1i  1 + 1i −1 + 1i  1 +1i 0 0 1 1 1 0 0 0 −1 − 1i  1 + 1i  1 − 1i −1 − 1i 0 0 1 1 1 0 1 0 −1 −1i  1 + 1i  1 − 1i  1 − 1i 0 0 1 1 1 1 0 1 −1 − 1i  1 + 1i  1 + 1i −1 +1i 0 0 1 1 1 1 1 1 −1 − 1i  1 + 1i  1 + 1i  1 + 1i 0 1 0 0 0 0 0 1 −1 +1i −1 − 1i −1 − 1i −1 + 1i 0 1 0 0 0 0 1 1 −1 + 1i −1 − 1i −1 − 1i  1 +1i 0 1 0 0 0 1 0 1 −1 + 1i −1 − 1i −1 + 1i −1 + 1i 0 1 0 0 0 1 1 1 −1 +1i −1 − 1i −1 + 1i  1 + 1i 0 1 0 0 1 0 0 0 −1 + 1i −1 − 1i  1 − 1i −1 −1i 0 1 0 0 1 0 1 0 −1 + 1i −1 − 1i  1 − 1i  1 − 1i 0 1 0 0 1 1 0 0 −1 +1i −1 − 1i  1 + 1i −1 − 1i 0 1 0 0 1 1 1 0 −1 + 1i −1 − 1i  1 + 1i  1 −1i 0 1 0 1 0 0 0 1 −1 + 1i −1 + 1i −1 − 1i −1 + 1i 0 1 0 1 0 0 1 1 −1 +1i −1 + 1i −1 − 1i  1 + 1i 0 1 0 1 0 1 0 0 −1 + 1i −1 + 1i −1 + 1i −1 −1i 0 1 0 1 0 1 1 0 −1 + 1i −1 + 1i −1 + 1i  1 − 1i 0 1 0 1 1 0 0 1 −1 +1i −1 + 1i  1 − 1i −1 + 1i 0 1 0 1 1 0 1 1 −1 + 1i −1 + 1i  1 − 1i  1 +1i 0 1 0 1 1 1 0 0 −1 + 1i −1 + 1i  1 + 1i −1 − 1i 0 1 0 1 1 1 1 0 −1 +1i −1 + 1i  1 + 1i  1 − 1i 0 1 1 0 0 0 0 0 −1 + 1i  1 − 1i −1 − 1i −1 −1i 0 1 1 0 0 0 1 0 −1 + 1i  1 − 1i −1 − 1i  1 − 1i 0 1 1 0 0 1 0 1 −1 +1i  1 − 1i −1 + 1i −1 + 1i 0 1 1 0 0 1 1 1 −1 + 1i  1 − 1i −1 + 1i  1 +1i 0 1 1 0 1 0 0 0 −1 + 1i  1 − 1i  1 − 1i −1 − 1i 0 1 1 0 1 0 1 0 −1 +1i  1 − 1i  1 − 1i  1 − 1i 0 1 1 0 1 1 0 1 −1 + 1i  1 − 1i  1 + 1i −1 +1i 0 1 1 0 1 1 1 1 −1 + 1i  1 − 1i  1 + 1i  1 + 1i 0 1 1 1 0 0 0 0 −1 +1i  1 + 1i −1 − 1i −1 − 1i 0 1 1 1 0 0 1 0 −1 + 1i  1 + 1i −1 − 1i  1 −1i 0 1 1 1 0 1 0 0 −1 + 1i  1 + 1i −1 + 1i −1 − 1i 0 1 1 1 0 1 1 0 −1 +1i  1 + 1i −1 + 1i  1 − 1i 0 1 1 1 1 0 0 1 −1 + 1i  1 + 1i  1 − 1i −1 +1i 0 1 1 1 1 0 1 1 −1 + 1i  1 + 1i  1 − 1i  1 + 1i 0 1 1 1 1 1 0 1 −1 +1i  1 + 1i  1 + 1i −1 + 1i 0 1 1 1 1 1 1 1 −1 + 1i  1 + 1i  1 + 1i  1 +1i 1 0 0 0 0 0 0 0  1 − 1i −1 − 1i −1 − 1i −1 − 1i 1 0 0 0 0 0 1 0  1 −1i −1 − 1i −1 − 1i  1 − 1i 1 0 0 0 0 1 0 0  1 − 1i −1 − 1i −1 + 1i −1 −1i 1 0 0 0 0 1 1 0  1 − 1i −1 − 1i −1 + 1i  1 − 1i 1 0 0 0 1 0 0 1  1 −1i −1 − 1i  1 − 1i −1 + 1i 1 0 0 0 1 0 1 1  1 − 1i −1 − 1i  1 − 1i  1 +1i 1 0 0 0 1 1 0 1  1 − 1i −1 − 1i  1 + 1i −1 + 1i 1 0 0 0 1 1 1 1  1 −1i −1 − 1i  1 + 1i  1 + 1i 1 0 0 1 0 0 0 0  1 − 1i −1 + 1i −1 − 1i −1 −1i 1 0 0 1 0 0 1 0  1 − 1i −1 + 1i −1 − 1i  1 − 1i 1 0 0 1 0 1 0 1  1 −1i −1 + 1i −1 + 1i −1 + 1i 1 0 0 1 0 1 1 1  1 − 1i −1 + 1i −1 + 1i  1 +1i 1 0 0 1 1 0 0 0  1 − 1i −1 + 1i  1 − 1i −1 − 1i 1 0 0 1 1 0 1 0  1 −1i −1 + 1i  1 − 1i  1 − 1i 1 0 0 1 1 1 0 1  1 − 1i −1 + 1i  1 + 1i −1 +1i 1 0 0 1 1 1 1 1  1 − 1i −1 + 1i  1 + 1i  1 + 1i 1 0 1 0 0 0 0 1  1 −1i  1 − 1i −1 − 1i −1 + 1i 1 0 1 0 0 0 1 1  1 − 1i  1 − 1i −1 − 1i  1 +1i 1 0 1 0 0 1 0 0  1 − 1i  1 − 1i −1 + 1i −1 − 1i 1 0 1 0 0 1 1 0  1 −1i  1 − 1i −1 + 1i  1 − 1i 1 0 1 0 1 0 0 1  1 − 1i  1 − 1i  1 − 1i −1 +1i 1 0 1 0 1 0 1 1  1 − 1i  1 − 1i  1 − 1i  1 + 1i 1 0 1 0 1 1 0 0  1 −1i  1 − 1i  1 + 1i −1 − 1i 1 0 1 0 1 1 1 0  1 − 1i  1 − 1i  1 + 1i  1 −1i 1 0 1 1 0 0 0 1  1 − 1i  1 + 1i −1 − 1i −1 + 1i 1 0 1 1 0 0 1 1  1 −1i  1 + 1i −1 − 1i  1 + 1i 1 0 1 1 0 1 0 1  1 − 1i  1 + 1i −1 + 1i −1 +1i 1 0 1 1 0 1 1 1  1 − 1i  1 + 1i −1 + 1i  1 + 1i 1 0 1 1 1 0 0 0  1 −1i  1 + 1i  1 − 1i −1 − 1i 1 0 1 1 1 0 1 0  1 − 1i  1 + 1i  1 − 1i  1 −1i 1 0 1 1 1 1 0 0  1 − 1i  1 + 1i  1 + 1i −1 − 1i 1 0 1 1 1 1 1 0  1 −1i  1 + 1i  1 + 1i  1 − 1i 1 1 0 0 0 0 0 0  1 + 1i −1 − 1i −1 − 1i −1 −1i 1 1 0 0 0 0 1 0  1 + 1i −1 − 1i −1 − 1i  1 − 1i 1 1 0 0 0 1 0 1  1 +1i −1 − 1i −1 + 1i −1 + 1i 1 1 0 0 0 1 1 1  1 + 1i −1 − 1i −1 + 1i  1 +1i 1 1 0 0 1 0 0 0  1 + 1i −1 − 1i  1 − 1i −1 − 1i 1 1 0 0 1 0 1 0  1 +1i −1 − 1i  1 − 1i  1 − 1i 1 1 0 0 1 1 0 1  1 + 1i −1 − 1i  1 + 1i −1 +1i 1 1 0 0 1 1 1 1  1 + 1i −1 − 1i  1 + 1i  1 + 1i 1 1 0 1 0 0 0 1  1 +1i −1 + 1i −1 − 1i −1 + 1i 1 1 0 1 0 0 1 1  1 + 1i −1 + 1i −1 − 1i  1 +1i 1 1 0 1 0 1 0 1  1 + 1i −1 + 1i −1 + 1i −1 + 1i 1 1 0 1 0 1 1 1  1 +1i −1 + 1i −1 + 1i  1 + 1i 1 1 0 1 1 0 0 0  1 + 1i −1 + 1i  1 − 1i −1 −1i 1 1 0 1 1 0 1 0  1 + 1i −1 + 1i  1 − 1i  1 − 1i 1 1 0 1 1 1 0 0  1 +1i −1 + 1i  1 + 1i −1 − 1i 1 1 0 1 1 1 1 0  1 + 1i −1 + 1i  1 + 1i  1 −1i 1 1 1 0 0 0 0 0  1 + 1i  1 − 1i −1 − 1i −1 − 1i 1 1 1 0 0 0 1 0  1 +1i  1 − 1i −1 − 1i  1 − 1i 1 1 1 0 0 1 0 0  1 + 1i  1 − 1i −1 + 1i −1 −1i 1 1 1 0 0 1 1 0  1 + 1i  1 − 1i −1 + 1i  1 − 1i 1 1 1 0 1 0 0 1  1 +1i  1 − 1i  1 − 1i −1 + 1i 1 1 1 0 1 0 1 1  1 + 1i  1 − 1i  1 − 1i  1 +1i 1 1 1 0 1 1 0 1  1 + 1i  1 − 1i  1 + 1i −1 + 1i 1 1 1 0 1 1 1 1  1 +1i  1 − 1i  1 + 1i  1 + 1i 1 1 1 1 0 0 0 1  1 + 1i  1 + 1i −1 − 1i −1 +1i 1 1 1 1 0 0 1 1  1 + 1i  1 + 1i −1 − 1i  1 + 1i 1 1 1 1 0 1 0 0  1 +1i  1 + 1i −1 + 1i −1 − 1i 1 1 1 1 0 1 1 0  1 + 1i  1 + 1i −1 + 1i  1 −1i 1 1 1 1 1 0 0 1  1 + 1i  1 + 1i  1 − 1i −1 + 1i 1 1 1 1 1 0 1 1  1 +1i  1 + 1i  1 − 1i  1 + 1i 1 1 1 1 1 1 0 0  1 + 1i  1 + 1i  1 + 1i −1 −1i 1 1 1 1 1 1 1 0  1 + 1i  1 + 1i  1 + 1i  1 − 1i

The above-given table and formulas define the modulation formataccording to the previous implementation forms up to symmetries andchange of labels, which lead to formats with the same performance.

According to the above implementation forms, the modulation format atthe spectral efficiency of 3.5 bits/interval in 8D is realized, whichhas better linear performance than the conventional solution with thesame spectral efficiency. In particular, the modulation format hasbetter linear and nonlinear performance. This is due to the fact thatthe constellation of the four symbols used in the first aspect has ahigher Euclidian distance, which gives better linear performance.

In a further implementation form of the first aspect, the encoder isconfigured to multiply each symbol selected from the set of four symbols−1−1i, −1+1i, 1−1i and 1+1i by a real number.

Multiplying all symbols by a real number still obtains a modulationformat with the desired properties. In particular, multiplying a QPSKsymbol with a real number does not change the polarization states.Depending on the implementation, the multiplication may not have aneffect, or may simply change the transmission power.

A second aspect of the disclosure provides an optical transmissionsystem, comprising the optical transmitter according to the first aspectas such or any implementation form of the first aspect, and an opticalreceiver for receiving the data signal, wherein the optical receiver isconfigured to receive and decode the modulated carrier waves of theoptical carrier to obtain the data signal.

The optical transmission system of the second aspect achieves alladvantages and effects of the optical transmitter of the first aspect.

A third aspect of the disclosure provides a method of opticallytransmitting a data signal, comprising encoding the data signal byselecting based on a bit sequence a first symbol and a second symbolfrom a set of four symbols for each one of at least two transmissiontime slots, and using in each transmission time slot the first symbol tomodulate a first carrier wave and the second symbol to modulate a secondcarrier wave, and transmitting the two carrier waves over orthogonalpolarizations of an optical carrier, wherein symbols in consecutivetransmission time slots have non-identical polarization states.

In an implementation form of the third aspect, the method comprisesselecting the symbols from a QPSK base constellation.

In a further implementation form of the third aspect, the methodcomprises modulating an In-Phase and a Quadrature component of eachcarrier wave.

In a further implementation form of the third aspect, symbols in atleast a subset of consecutive transmission time slots have anti-parallelpolarization states.

In a further implementation form of the third aspect, the method furthercomprises generating the bit sequence based on the data signal, and thedata signal comprises less bits than the bit sequence.

In a further implementation form of the third aspect, the methodcomprises performing at least one arithmetic operation based on at leasttwo bits of the data signal to obtain at least one overhead bit, and togenerate the bit sequence based on the bits of the data signal and atleast one overhead bit.

In a further implementation form of the third aspect, the methodcomprises transmitting the data signal with a spectral efficiency of 2.5bits per transmission time slot.

In a further implementation form of the third aspect, the symbolpolarization states in each transmission time slot take one of at leastfour distinct polarizations states.

In a further implementation form of the third aspect, the data signalhas five bits b1 . . . b5, and the method comprises generating the bitsequence having eight bits b1 . . . b5, b1′, b2′, b3′, wherein threeoverhead bits b1′, b2′, b3′ are generated according tob1′=b3XORb4XORb5b2′= b2XORb4XORb5b3′= b1XORb4XORb5

In a further implementation form of the third aspect, for twoconsecutive transmission time slots T1 and T2, for two orthogonalpolarizations X and Y of the optical carrier, and for a set of four QPSKsymbols denoted −1−1i, −1+1i, 1−1i and 1+1i, the method comprisesselecting the symbols based on the data signal according to thefollowing labelling:

Labelling (from left to right) 5 bits b1 . . . b5, and 3 overhead Timeslot T₁ Time slot T₂ bits b1′, b2′, b3′ X polarization Y polarization Xpolarization Y polarization 0 0 0 0 0 0 1 1 −1 − 1i −1 − 1i −1 − 1i  1 +1i 0 0 0 0 1 1 0 0 −1 − 1i −1 − 1i  1 + 1i −1 − 1i 0 0 0 1 0 1 0 0 −1 −1i −1 + 1i −1 + 1i −1 − 1i 0 0 0 1 1 0 1 1 −1 − 1i −1 + 1i  1 − 1i  1 +1i 0 0 1 0 0 1 1 1 −1 − 1i  1 − 1i −1 + 1i  1 + 1i 0 0 1 0 1 0 0 0 −1 −1i  1 − 1i  1 − 1i −1 − 1i 0 0 1 1 0 0 0 0 −1 − 1i  1 + 1i −1 − 1i −1 −1i 0 0 1 1 1 1 1 1 −1 − 1i  1 + 1i  1 + 1i  1 + 1i 0 1 0 0 0 0 0 1 −1 +1i −1 − 1i −1 − 1i −1 + 1i 0 1 0 0 1 1 1 0 −1 + 1i −1 − 1i  1 + 1i  1 −1i 0 1 0 1 0 1 1 0 −1 + 1i −1 + 1i −1 + 1i  1 − 1i 0 1 0 1 1 0 0 1 −1 +1i −1 + 1i  1 − 1i −1 + 1i 0 1 1 0 0 1 0 1 −1 + 1i  1 − 1i −1 + 1i −1 +1i 0 1 1 0 1 0 1 0 −1 + 1i  1 − 1i  1 − 1i  1 − 1i 0 1 1 1 0 0 1 0 −1 +1i  1 + 1i −1 − 1i  1 − 1i 0 1 1 1 1 1 0 1 −1 + 1i  1 + 1i  1 + 1i −1 +1i 1 0 0 0 0 0 1 0  1 − 1i −1 − 1i −1 − 1i  1 − 1i 1 0 0 0 1 1 0 1  1 −1i −1 − 1i  1 + 1i −1 + 1i 1 0 0 1 0 1 0 1  1 − 1i −1 + 1i −1 + 1i −1 +1i 1 0 0 1 1 0 1 0  1 − 1i −1 + 1i  1 − 1i  1 − 1i 1 0 1 0 0 1 1 0  1 −1i  1 − 1i −1 + 1i  1 − 1i 1 0 1 0 1 0 0 1  1 − 1i  1 − 1i  1 − 1i −1 +1i 1 0 1 1 0 0 0 1  1 − 1i  1 + 1i −1 − 1i −1 + 1i 1 0 1 1 1 1 1 0  1 −1i  1 + 1i  1 + 1i  1 − 1i 1 1 0 0 0 0 0 0  1 + 1i −1 − 1i −1 − 1i −1 −1i 1 1 0 0 1 1 1 1  1 + 1i −1 − 1i  1 + 1i  1 + 1i 1 1 0 1 0 1 1 1  1 +1i −1 + 1i −1 + 1i  1 + 1i 1 1 0 1 1 0 0 0  1 + 1i −1 + 1i  1 − 1i −1 −1i 1 1 1 0 0 1 0 0  1 + 1i  1 − 1i −1 + 1i −1 − 1i 1 1 1 0 1 0 1 1  1 +1i  1 − 1i  1 − 1i  1 + 1i 1 1 1 1 0 0 1 1  1 + 1i  1 + 1i −1 − 1i  1 +1i 1 1 1 1 1 1 0 0  1 + 1i  1 + 1i  1 + 1i −1 − 1i

In a further implementation form of the third aspect, the methodcomprises transmitting the data signal with a spectral efficiency of 3.5bits per transmission time slot.

In a further implementation form of the third aspect, symbols in atleast a subset of consecutive transmission time slots have orthogonalpolarization states.

In a further implementation form of the third aspect, the data signalhas seven bits b1 . . . b7, and the method comprises generating the bitsequence having eight bits b1 . . . b7, b′, wherein the overhead bit b′is generated according to:b′=b1XORb4XORb6XOR(b1ANDb3)XOR(b1ANDb4)XOR(b1ANDb5)XOR(b1ANDb6)XOR(b2ANDb3)XOR(b2ANDb4)XOR(b2ANDb5)XOR(b2ANDb6)XOR(b3ANDb5)XOR(b3ANDb6)XOR(b4ANDb5)XOR(b4ANDb6)

In a further implementation form of the third aspect, for twoconsecutive transmission time slots T1 and T2, for two orthogonalpolarizations X and Y of the optical carrier, and for a set of four QPSKsymbols denoted −1−1i, −1+1i, 1−1i and 1+1i, the method comprisesselecting the symbols based on the data signal according to thefollowing labelling:

Labelling (from left to right) 7 bits b1 . . . b7, Time slot T₁ Timeslot T₂ and 1 overhead bit b′ X polarization Y polarization Xpolarization Y polarization 0 0 0 0 0 0 0 1 −1 − 1i −1 − 1i −1 − 1i −1 +1i 0 0 0 0 0 0 1 1 −1 − 1i −1 − 1i −1 − 1i  1 + 1i 0 0 0 0 0 1 0 0 −1 −1i −1 − 1i −1 + 1i −1 − 1i 0 0 0 0 0 1 1 0 −1 − 1i −1 − 1i −1 + 1i  1 −1i 0 0 0 0 1 0 0 1 −1 − 1i −1 − 1i  1 − 1i −1 + 1i 0 0 0 0 1 0 1 1 −1 −1i −1 − 1i  1 − 1i  1 + 1i 0 0 0 0 1 1 0 0 −1 − 1i −1 − 1i  1 + 1i −1 −1i 0 0 0 0 1 1 1 0 −1 − 1i −1 − 1i  1 + 1i  1 − 1i 0 0 0 1 0 0 0 0 −1 −1i −1 + 1i −1 − 1i −1 − 1i 0 0 0 1 0 0 1 0 −1 − 1i −1 + 1i −1 − 1i  1 −1i 0 0 0 1 0 1 0 0 −1 − 1i −1 + 1i −1 + 1i −1 − 1i 0 0 0 1 0 1 1 0 −1 −1i −1 + 1i −1 + 1i  1 − 1i 0 0 0 1 1 0 0 1 −1 − 1i −1 + 1i  1 − 1i −1 +1i 0 0 0 1 1 0 1 1 −1 − 1i −1 + 1i  1 − 1i  1 + 1i 0 0 0 1 1 1 0 1 −1 −1i −1 + 1i  1 + 1i −1 + 1i 0 0 0 1 1 1 1 1 −1 − 1i −1 + 1i  1 + 1i  1 +1i 0 0 1 0 0 0 0 1 −1 − 1i  1 − 1i −1 − 1i −1 + 1i 0 0 1 0 0 0 1 1 −1 −1i  1 − 1i −1 − 1i  1 + 1i 0 0 1 0 0 1 0 1 −1 − 1i  1 − 1i −1 + 1i −1 +1i 0 0 1 0 0 1 1 1 −1 − 1i  1 − 1i −1 + 1i  1 + 1i 0 0 1 0 1 0 0 0 −1 −1i  1 − 1i  1 − 1i −1 − 1i 0 0 1 0 1 0 1 0 −1 − 1i  1 − 1i  1 − 1i  1 −1i 0 0 1 0 1 1 0 0 −1 − 1i  1 − 1i  1 + 1i −1 − 1i 0 0 1 0 1 1 1 0 −1 −1i  1 − 1i  1 + 1i  1 − 1i 0 0 1 1 0 0 0 0 −1 − 1i  1 + 1i −1 − 1i −1 −1i 0 0 1 1 0 0 1 0 −1 − 1i  1 + 1i −1 − 1i  1 − 1i 0 0 1 1 0 1 0 1 −1 −1i  1 + 1i −1 + 1i −1 + 1i 0 0 1 1 0 1 1 1 −1 − 1i  1 + 1i −1 + 1i  1 +1i 0 0 1 1 1 0 0 0 −1 − 1i  1 + 1i  1 − 1i −1 − 1i 0 0 1 1 1 0 1 0 −1 −1i  1 + 1i  1 − 1i  1 − 1i 0 0 1 1 1 1 0 1 −1 − 1i  1 + 1i  1 + 1i −1 +1i 0 0 1 1 1 1 1 1 −1 − 1i  1 + 1i  1 + 1i  1 + 1i 0 1 0 0 0 0 0 1 −1 +1i −1 − 1i −1 − 1i −1 + 1i 0 1 0 0 0 0 1 1 −1 + 1i −1 − 1i −1 − 1i  1 +1i 0 1 0 0 0 1 0 1 −1 + 1i −1 − 1i −1 + 1i −1 + 1i 0 1 0 0 0 1 1 1 −1 +1i −1 − 1i −1 + 1i  1 + 1i 0 1 0 0 1 0 0 0 −1 + 1i −1 − 1i  1 − 1i −1 −1i 0 1 0 0 1 0 1 0 −1 + 1i −1 − 1i  1 − 1i  1 − 1i 0 1 0 0 1 1 0 0 −1 +1i −1 − 1i  1 + 1i −1 − 1i 0 1 0 0 1 1 1 0 −1 + 1i −1 − 1i  1 + 1i  1 −1i 0 1 0 1 0 0 0 1 −1 + 1i −1 + 1i − 1 − 1i −1 + 1i 0 1 0 1 0 0 1 1 −1 +1i −1 + 1i − 1 − 1i  1 + 1i 0 1 0 1 0 1 0 0 −1 + 1i −1 + 1i − 1 + 1i −1− 1i 0 1 0 1 0 1 1 0 −1 + 1i −1 + 1i − 1 + 1i  1 − 1i 0 1 0 1 1 0 0 1−1 + 1i −1 + 1i  1 − 1i −1 + 1i 0 1 0 1 1 0 1 1 −1 + 1i −1 + 1i  1 − 1i 1 + 1i 0 1 0 1 1 1 0 0 −1 + 1i −1 + 1i  1 + 1i −1 − 1i 0 1 0 1 1 1 1 0−1 + 1i −1 + 1i  1 + 1i  1 − 1i 0 1 1 0 0 0 0 0 −1 + 1i  1 − 1i −1 − 1i−1 − 1i 0 1 1 0 0 0 1 0 −1 + 1i  1 − 1i −1 − 1i  1 − 1i 0 1 1 0 0 1 0 1−1 + 1i  1 − 1i −1 + 1i −1 + 1i 0 1 1 0 0 1 1 1 −1 + 1i  1 − 1i −1 + 1i 1 + 1i 0 1 1 0 1 0 0 0 −1 + 1i  1 − 1i  1 − 1i −1 − 1i 0 1 1 0 1 0 1 0−1 + 1i  1 − 1i  1 − 1i  1 − 1i 0 1 1 0 1 1 0 1 −1 + 1i  1 − 1i  1 + 1i−1 + 1i 0 1 1 0 1 1 1 1 −1 + 1i  1 − 1i  1 + 1i  1 + 1i 0 1 1 1 0 0 0 0−1 + 1i  1 + 1i −1 − 1i −1 − 1i 0 1 1 1 0 0 1 0 −1 + 1i  1 + 1i −1 − 1i 1 − 1i 0 1 1 1 0 1 0 0 −1 + 1i  1 + 1i −1 + 1i −1 − 1i 0 1 1 1 0 1 1 0−1 + 1i  1 + 1i −1 + 1i  1 − 1i 0 1 1 1 1 0 0 1 −1 + 1i  1 + 1i  1 − 1i−1 + 1i 0 1 1 1 1 0 1 1 −1 + 1i  1 + 1i  1 − 1i  1 + 1i 0 1 1 1 1 1 0 1−1 + 1i  1 + 1i  1 + 1i −1 + 1i 0 1 1 1 1 1 1 1 −1 + 1i  1 + 1i  1 + 1i 1 + 1i 1 0 0 0 0 0 0 0  1 − 1i −1 − 1i −1 − 1i −1 − 1i 1 0 0 0 0 0 1 0 1 − 1i −1 − 1i −1 − 1i  1 − 1i 1 0 0 0 0 1 0 0  1 − 1i −1 − 1i −1 + 1i−1 − 1i 1 0 0 0 0 1 1 0  1 − 1i −1 − 1i −1 + 1i  1 − 1i 1 0 0 0 1 0 0 1 1 − 1i −1 − 1i  1 − 1i −1 + 1i 1 0 0 0 1 0 1 1  1 − 1i −1 − 1i  1 − 1i 1 + 1i 1 0 0 0 1 1 0 1  1 − 1i −1 − 1i  1 + 1i −1 + 1i 1 0 0 0 1 1 1 1 1 − 1i −1 − 1i  1 + 1i  1 + 1i 1 0 0 1 0 0 0 0  1 − 1i −1 + 1i −1 − 1i−1 − 1i 1 0 0 1 0 0 1 0  1 − 1i −1 + 1i −1 − 1i  1 − 1i 1 0 0 1 0 1 0 1 1 − 1i −1 + 1i −1 + 1i −1 + 1i 1 0 0 1 0 1 1 1  1 − 1i −1 + 1i −1 + 1i 1 + 1i 1 0 0 1 1 0 0 0  1 − 1i −1 + 1i  1 − 1i −1 − 1i 1 0 0 1 1 0 1 0 1 − 1i −1 + 1i  1 − 1i  1 − 1i 1 0 0 1 1 1 0 1  1 − 1i −1 + 1i  1 + 1i−1 + 1i 1 0 0 1 1 1 1 1  1 − 1i −1 + 1i  1 + 1i  1 + 1i 1 0 1 0 0 0 0 1 1 − 1i  1 − 1i −1 − 1i −1 + 1i 1 0 1 0 0 0 1 1  1 − 1i  1 − 1i −1 − 1i 1 + 1i 1 0 1 0 0 1 0 0  1 − 1i  1 − 1i −1 + 1i −1 − 1i 1 0 1 0 0 1 1 0 1 − 1i  1 − 1i −1 + 1i  1 − 1i 1 0 1 0 1 0 0 1  1 − 1i  1 − 1i  1 − 1i−1 + 1i 1 0 1 0 1 0 1 1  1 − 1i  1 − 1i  1 − 1i  1 + 1i 1 0 1 0 1 1 0 0 1 − 1i  1 − 1i  1 + 1i −1 − 1i 1 0 1 0 1 1 1 0  1 − 1i  1 − 1i  1 + 1i 1 − 1i 1 0 1 1 0 0 0 1  1 − 1i  1 + 1i −1 − 1i −1 + 1i 1 0 1 1 0 0 1 1 1 − 1i  1 + 1i −1 − 1i  1 + 1i 1 0 1 1 0 1 0 1  1 − 1i  1 + 1i −1 + 1i−1 + 1i 1 0 1 1 0 1 1 1  1 − 1i  1 + 1i −1 + 1i  1 + 1i 1 0 1 1 1 0 0 0 1 − 1i  1 + 1i  1 − 1i −1 − 1i 1 0 1 1 1 0 1 0  1 − 1i  1 + 1i  1 − 1i 1 − 1i 1 0 1 1 1 1 0 0  1 − 1i  1 + 1i  1 + 1i −1 − 1i 1 0 1 1 1 1 1 0 1 − 1i  1 + 1i  1 + 1i  1 − 1i 1 1 0 0 0 0 0 0  1 + 1i −1 − 1i −1 − 1i−1 − 1i 1 1 0 0 0 0 1 0  1 + 1i −1 − 1i −1 − 1i  1 − 1i 1 1 0 0 0 1 0 1 1 + 1i −1 − 1i −1 + 1i −1 + 1i 1 1 0 0 0 1 1 1  1 + 1i −1 − 1i −1 + 1i 1 + 1i 1 1 0 0 1 0 0 0  1 + 1i −1 − 1i  1 − 1i −1 − 1i 1 1 0 0 1 0 1 0 1 + 1i −1 − 1i  1 − 1i  1 − 1i 1 1 0 0 1 1 0 1  1 + 1i −1 − 1i  1 + 1i−1 + 1i 1 1 0 0 1 1 1 1  1 + 1i −1 − 1i  1 + 1i  1 + 1i 1 1 0 1 0 0 0 1 1 + 1i −1 + 1i −1 − 1i −1 + 1i 1 1 0 1 0 0 1 1  1 + 1i −1 + 1i −1 − 1i 1 + 1i 1 1 0 1 0 1 0 1  1 + 1i −1 + 1i −1 + 1i −1 + 1i 1 1 0 1 0 1 1 1 1 + 1i −1 + 1i −1 + 1i  1 + 1i 1 1 0 1 1 0 0 0  1 + 1i −1 + 1i  1 − 1i−1 − 1i 1 1 0 1 1 0 1 0  1 + 1i −1 + 1i  1 − 1i  1 − 1i 1 1 0 1 1 1 0 0 1 + 1i −1 + 1i  1 + 1i −1 − 1i 1 1 0 1 1 1 1 0  1 + 1i −1 + 1i  1 + 1i 1 − 1i 1 1 1 0 0 0 0 0  1 + 1i  1 − 1i −1 − 1i −1 − 1i 1 1 1 0 0 0 1 0 1 + 1i  1 − 1i −1 − 1i  1 − 1i 1 1 1 0 0 1 0 0  1 + 1i  1 − 1i −1 + 1i−1 − 1i 1 1 1 0 0 1 1 0  1 + 1i  1 − 1i −1 + 1i  1 − 1i 1 1 1 0 1 0 0 1 1 + 1i  1 − 1i  1 − 1i −1 + 1i 1 1 1 0 1 0 1 1  1 + 1i  1 − 1i  1 − 1i 1 + 1i 1 1 1 0 1 1 0 1  1 + 1i  1 − 1i  1 + 1i −1 + 1i 1 1 1 0 1 1 1 1 1 + 1i  1 − 1i  1 + 1i  1 + 1i 1 1 1 1 0 0 0 1  1 + 1i  1 + 1i −1 − 1i−1 + 1i 1 1 1 1 0 0 1 1  1 + 1i  1 + 1i −1 − 1i  1 + 1i 1 1 1 1 0 1 0 0 1 + 1i  1 + 1i −1 + 1i −1 − 1i 1 1 1 1 0 1 1 0  1 + 1i  1 + 1i −1 + 1i 1 − 1i 1 1 1 1 1 0 0 1  1 + 1i  1 + 1i  1 − 1i −1 + 1i 1 1 1 1 1 0 1 1 1 + 1i  1 + 1i  1 − 1i  1 + 1i 1 1 1 1 1 1 0 0  1 + 1i  1 + 1i  1 + 1i−1 − 1i 1 1 1 1 1 1 1 0  1 + 1i  1 + 1i  1 + 1i  1 − 1i

In a further implementation form of the third aspect, the methodcomprises multiplying each symbol selected from the set of four symbols−1−1i, −1+1i, 1−1i and 1+1i by a real number.

With the method of the third aspect and its implementation forms, alladvantages and effects of the optical transmitter of the first aspectand its respective implementation forms are achieved. The method mayfurther comprise a step of receiving, demodulating and decoding the datasignal at a receiver side.

It has to be noted that all devices, elements, units and means describedin the present disclosure could be implemented in the software orhardware elements or any kind of combination thereof. All steps whichare performed by the various entities described in the presentapplication as well as the functionalities described to be performed bythe various entities are intended to mean that the respective entity isadapted to or configured to perform the respective steps andfunctionalities.

Even if, in the following description of exemplary embodiments, aspecific functionality or step to be performed by external entities isnot reflected in the description of a specific detailed element of thatentity which performs that specific step or functionality, it should beclear for a skilled person that these methods and functionalities can beimplemented in respective software or hardware elements, or in anypossible kind of combination of such elements.

BRIEF DESCRIPTION OF THE DRAWINGS

The above described aspects and implementation forms of the disclosurewill be explained in the following description of exemplary embodimentsin relation to the enclosed drawings, in which:

FIG. 1 shows an optical transmitter according to an exemplary embodimentof the present disclosure.

FIG. 2 shows a set of four symbols, in particular a QPSK constellationand labelling.

FIG. 3 shows an optical transmission system according to an exemplaryembodiment of the present disclosure including an optical transmitteraccording to an embodiment of the present disclosure.

FIG. 4 shows an example encoder of an optical transmitter according toan exemplary embodiment of the present disclosure.

FIG. 5 shows a method according to an exemplary embodiment of thepresent disclosure.

DETAILED DESCRIPTION

The disclosure presents an optical transmitter 100, an opticaltransmission system, and a method 500, which use modulation formatsobtained by set-partitioning of a constellation whose projection onto aninterval is the PDM-QPSK constellation. The following constraints wereapplied for designing the modulation formats:

-   -   Symbols must not have identical (parallel) states of        polarization in two consecutive time slots.    -   If possible, symbols have opposite (antiparallel) states of        polarization in two consecutive time slots.    -   If there are not enough symbols to reach the desired spectral        efficiency, then missing symbols are chosen from the set of        symbols with the polarization alternating property (not        identical polarization states).    -   The overall set of symbols are chosen to have preferably a high        symmetry. More specifically, for each symbol of a given        modulation format, the Euclidean distances to its neighbors is        preferably chosen to be the highest possible one, with respect        to what the whole PDM-QPSK constellation offers.

The modulation formats used by the optical transmitter 100, thetransmission system, and the method 500 according to embodiments of thepresent disclosure, respectively, differ from the known modulationformats (at the same spectral efficiency) at least in that:

-   -   The modulation formats are derived from the same base        constellation of a set of four symbols.    -   The modulation formats have the same modulus in each of the        dimensions separately (this is in fact a consequence of the        previous point).    -   The modulation formats may contain symbols with the polarization        alternating property.    -   The modulation formats have at least four distinct polarization        states.

FIG. 1 shows an optical transmitter 100 according to an exemplaryembodiment of the present disclosure. The optical transmitter 100 isconfigured to transmit a data signal 101, which comprises a sequence ofbits, over an optical link like an optical carrier 104. The opticaltransmitter 100 comprises an encoder 102 for encoding the data signal101, and a modulator 103 for modulating and transmitting the data signal101 on the optical carrier 104. For instance, the optical carrier 104may be an optical fiber.

In particular, the encoder 102 is configured to encode the data signal101 by selecting, based on a bit sequence, a first symbol and a secondsymbol from a set 200 of four symbols 201-204 (see FIG. 2 ), for eachone of at least two transmission time slots. Notably, the bit sequencemay be the bits of the data signal 101 itself or may be a bit sequence401 derived from the data signal 101.

The modulator 103 is configured to use, in each transmission time slot,the first symbol to modulate a first carrier wave and the second symbolto modulate a second carrier wave. Further, the modulator 103 isconfigured to transmit the two carrier waves over orthogonalpolarizations of the optical carrier 104.

Symbols 201-204 in consecutive transmission time slots havenon-identical polarization states, i.e. they follow the above-described‘polarization alternating’ concept. The modulation symbols 201-204 for agiven polarization and time slot are preferably taken from the QPSKconstellation shown in FIG. 2 , providing the set 200 of four symbols201-204. The set 200 shown in FIG. 2 includes specifically the four QPSKsymbols denoted as −1−1i (symbol 201), −1+1i (symbol 203), 1−1i (symbol202) and 1+1i (symbols 204).

FIG. 3 shows an optical communication or transmission system accordingto an exemplary embodiment of the disclosure, in which the modulationformats may be implemented. The transmission system comprises an opticaltransmitter 100 according to an embodiment of the present disclosure,particularly the optical transmitter 100 shown in FIG. 1 , an optical(coherent) receiver 300 for receiving the data signal 101, and anoptical link (i.e. the optical carrier 104) between the transmitter 100and receiver 300. The optical receiver 300 is particularly configured toreceive and decode the modulated carrier waves of the optical carrier104, in order to obtain the data signal 101.

In the optical transmitter 100, the encoder 102 encodes the data signal101 and may generally generate a sequence of M drive signals from anM=4N-dimensional constellation, where N is the number of time slots. Thedrive signals from the encoder 102 in turn are used to drive themodulator 103, which modulates the respective dimensions onto the (X andY) polarizations of the optical carrier 104. The modulator 103 and alaser 105 of the optical transmitter 100 may be implemented usingdevices known in the art.

The optical receiver 300 is preferably a coherent receiver, whichincludes an optical beam splitter 301 to separate the received carrierwaves into X and Y polarizations. The two obtained signals are mixedseparately with a local oscillator 302 and a set of photodetectors 304detects the optical power of each of the mixed signals for eachpolarization generated by an optical hybrid 302. An analog to digitalconverter 305 (ADC) samples each current of the photodetectors 304. Thesample streams, which each represent one of the modulated dimensions ofthe optical carrier 104, are processed in a digital signal processing306 (DSP), which may include dispersion compensation and possibly otherequalization techniques and down-sampling. The processed sample streamis further processed in a decoder 307, such that samples correspondingto the same multi-dimensional constellation symbol 201-204 are processedjointly to recover the transmitted data signal 101. Specifically, thedecoder 307 in the receiver 300 performs the inverse operation of theencoder 102 in the transmitter 100.

The modulation formats detailed in the present disclosure areimplemented in the encoder 102 of the transmitter 100. An example forsuch an encoder 102 is shown in FIG. 4 . In the encoder 102, a number ofinput bits, namely information bits of the data signal 101 to betransmitter, is preferably mapped to a number of output bits of a bitsequence 401. This bit sequence 401 includes a number of overhead bits402 generated through arithmetic operations from the input bits. Theoutput bits are then mapped to multi-dimensional output symbols 201-204according to the labeling of the constellation points.

In the following, two refined embodiments are specifically described asexamples. These embodiments define the encoder 103 and decoder 307 andcorrespond to two different modulation formats in 8D with spectralefficiencies of 2.5 and 3.5 bit/transmission time slot, respectively.The two modulation formats both achieve the 8D through: I and Q, twoorthogonal polarizations referred to as X and Y, and two consecutivetime slots referred to as T₁ and T₂. For a given polarization and timeslot, the symbols 201-204 are chosen from the set of four symbols 200,preferably from the points in the I-Q-plane shown in FIG. 2 .

In the first exemplary embodiment, the modulation format is defined in8D: I, Q, polarization and two consecutive time-slots. The encoder 103(as shown in FIG. 4 ) maps 5 information bits of the data signal 101,referred to as [b₁, b₂, b₃, b₄, b₅] to eight output bits of the bitsequence 401. Three parity or overhead bits 402, referred to as b₁′, b₂′and b₃′, are defined using the following equations:b1′=b3XORb4XORb5b2′= b2XORb4XORb5b3′= b1XORb4XORb5

Thus, the set of [b₁ b₂ b₃ b₄ b₅ b₁′ b₂′ b₃′] is finally obtained. Thefirst two bits [b₁ b₂] are used to choose a symbol 201-204 from the set200 shown in FIG. 2 . These symbols 201-204 represent the 2 dimensions Iand Q of the X polarization on the time slot T₁. Then, the next two bits[b₃ b₄] are used to choose a symbol 201-204 from the set 200 of FIG. 2 ,which symbols 201-204 represents the 2 dimensions I and Q of the Ypolarization on the time slot T₁. The same approach is used for the bits[b₅ b₁′] and [b₂′ b₃′], respectively, selecting symbols 201-204 thatrepresent the 2 dimensions I and Q on the time slot T₂ of both X and Ypolarizations, respectively. At the end, a spectral efficiency of 2.5bits/transmission time slot (5 bits in 8 dimensions) is reached in thisembodiment, and all obtained symbols 201-204 are listed in the tablebelow. The labelling of the constellation points (the mapping frominformation bits of the data signal 101 to complex symbols 201-204)determines the linear channel performance.

Labelling (from left to right) 5 bits b1 . . . b5, and 3 overhead Timeslot T₁ Time slot T₂ bits b1′, b2′, b3′ X polarization Y polarization Xpolarization Y polarization 0 0 0 0 0 0 1 1 −1 − 1i −1 − 1i −1 − 1i  1 +1i 0 0 0 0 1 1 0 0 −1 − 1i −1 − 1i  1 + 1i −1 − 1i 0 0 0 1 0 1 0 0 −1 −1i −1 + 1i −1 + 1i −1 − 1i 0 0 0 1 1 0 1 1 −1 − 1i −1 + 1i  1 − 1i  1 +1i 0 0 1 0 0 1 1 1 −1 − 1i  1 − 1i −1 + 1i  1 + 1i 0 0 1 0 1 0 0 0 −1 −1i  1 − 1i  1 − 1i −1 − 1i 0 0 1 1 0 0 0 0 −1 − 1i  1 + 1i −1 − 1i −1 −1i 0 0 1 1 1 1 1 1 −1 − 1i  1 + 1i  1 + 1i  1 + 1i 0 1 0 0 0 0 0 1 −1 +1i −1 − 1i −1 − 1i −1 + 1i 0 1 0 0 1 1 1 0 −1 + 1i −1 − 1i  1 + 1i  1 −1i 0 1 0 1 0 1 1 0 −1 + 1i −1 + 1i −1 + 1i  1 − 1i 0 1 0 1 1 0 0 1 −1 +1i −1 + 1i  1 − 1i −1 + 1i 0 1 1 0 0 1 0 1 −1 + 1i  1 − 1i −1 + 1i −1 +1i 0 1 1 0 1 0 1 0 −1 + 1i  1 − 1i  1 − 1i  1 − 1i 0 1 1 1 0 0 1 0 −1 +1i  1 + 1i −1 − 1i  1 − 1i 0 1 1 1 1 1 0 1 −1 + 1i  1 + 1i  1 + 1i −1 +1i 1 0 0 0 0 0 1 0  1 − 1i −1 − 1i −1 − 1i  1 − 1i 1 0 0 0 1 1 0 1  1 −1i −1 − 1i  1 + 1i −1 + 1i 1 0 0 1 0 1 0 1  1 − 1i −1 + 1i −1 + 1i −1 +1i 1 0 0 1 1 0 1 0  1 − 1i −1 + 1i  1 − 1i  1 − 1i 1 0 1 0 0 1 1 0  1 −1i  1 − 1i −1 + 1i  1 − 1i 1 0 1 0 1 0 0 1  1 − 1i  1 − 1i  1 − 1i −1 +1i 1 0 1 1 0 0 0 1  1 − 1i  1 + 1i −1 − 1i −1 + 1i 1 0 1 1 1 1 1 0  1 −1i  1 + 1i  1 + 1i  1 − 1i 1 1 0 0 0 0 0 0  1 + 1i −1 − 1i −1 − 1i −1 −1i 1 1 0 0 1 1 1 1  1 + 1i −1 − 1i  1 + 1i  1 + 1i 1 1 0 1 0 1 1 1  1 +1i −1 + 1i −1 + 1i  1 + 1i 1 1 0 1 1 0 0 0  1 + 1i −1 + 1i  1 − 1i −1 −1i 1 1 1 0 0 1 0 0  1 + 1i  1 − 1i −1 + 1i −1 − 1i 1 1 1 0 1 0 1 1  1 +1i  1 − 1i  1 − 1i  1 + 1i 1 1 1 1 0 0 1 1  1 + 1i  1 + 1i −1 − 1i  1 +1i 1 1 1 1 1 1 0 0  1 + 1i  1 + 1i  1 + 1i −1 − 1i

These symbols 201-204 have an overall of 4 possible states ofpolarization with the condition that the state of polarization in T₂ isopposite to the one of T₁. The constellation has a high symmetry. Thestructure is such that the each constellation point has the same numberof neighbors. The neighbors are located at 4 different Euclideandistances, as shown on the following table.

Euclidean Distance Number of neighboring symbols 2.82 4 4 22 4.89 4 5.651

The above table shows that every point of the constellation has 4, 22, 4and 1 symbols at Euclidean distances of 2.82, 4, 4.89 and 5.65,respectively.

In the second exemplary embodiment, the modulation format is defined in8D: I, Q, polarization and two consecutive time slots. To map bits intosymbols 201-204, the following approach is used: from 7 information bitsof the data signal 101, referred to as [b₁, b₂, b₃, b₄, b₅, b₆, b₇], oneoverhead bit (b′) is obtained using the following equation:b1′=b1XORb4XORb6XOR(b1XORb2)AND(b3XORb4XORb5XORb6)XOR(b3XORb4)AND(b5XORb6)

Thus, the set of [b₁, b₂, b₃, b₄, b₅, b₆, b₇, b′] is obtained. Themapping is then done as follows. The first two bits [b₁ b₂] are used toselect a symbol 201-204 from the set 200 (QPSK constellation) shown inFIG. 2 . These symbols represents the 2 dimensions I and Q of the Xpolarization on the time slot T₁. Using the same approach, I and Qsymbols of Y polarization on T₁, X polarization on T₂ and finally Ypolarization on T₂ are obtained using the bits [b₃ b₄], [b₅ b₆] andfinally [b₇ b′], respectively. At the end, a spectral efficiency of 3.5bits/transmission time slot (7 bits in 8 dimensions) is reached, and allobtained symbols 201-204 are given in the following table. The labelling(mapping from bits to symbols 201-204) of each symbol 201-204 is givenin the table as well, because as mentioned before, different labellingmay result on different linear channel performance (except forequivalent constellations related by symmetry).

Labelling (from left to right) 7 bits b1 . . . b7, Time slot T₁ Timeslot T₂ and 1 overhead bit b′ X polarization Y polarization Xpolarization Y polarization 0 0 0 0 0 0 0 1 −1 − 1i −1 − 1i −1 − 1i −1 +1i 0 0 0 0 0 0 1 1 −1 − 1i −1 − 1i −1 − 1i  1 + 1i 0 0 0 0 0 1 0 0 −1 −1i −1 − 1i −1 + 1i −1 − 1i 0 0 0 0 0 1 1 0 −1 − 1i −1 − 1i −1 + 1i  1 −1i 0 0 0 0 1 0 0 1 −1 − 1i −1 − 1i  1 − 1i −1 + 1i 0 0 0 0 1 0 1 1 −1 −1i −1 − 1i  1 − 1i  1 + 1i 0 0 0 0 1 1 0 0 −1 − 1i −1 − 1i  1 + 1i −1 −1i 0 0 0 0 1 1 1 0 −1 − 1i −1 − 1i  1 + 1i  1 − 1i 0 0 0 1 0 0 0 0 −1 −1i −1 + 1i −1 − 1i −1 − 1i 0 0 0 1 0 0 1 0 −1 − 1i −1 + 1i −1 − 1i  1 −1i 0 0 0 1 0 1 0 0 −1 − 1i −1 + 1i −1 + 1i −1 − 1i 0 0 0 1 0 1 1 0 −1 −1i −1 + 1i −1 + 1i  1 − 1i 0 0 0 1 1 0 0 1 −1 − 1i −1 + 1i  1 − 1i −1 +1i 0 0 0 1 1 0 1 1 −1 − 1i −1 + 1i  1 − 1i  1 + 1i 0 0 0 1 1 1 0 1 −1 −1i −1 + 1i  1 + 1i −1 + 1i 0 0 0 1 1 1 1 1 −1 − 1i −1 + 1i  1 + 1i  1 +1i 0 0 1 0 0 0 0 1 −1 − 1i  1 − 1i −1 − 1i −1 + 1i 0 0 1 0 0 0 1 1 −1 −1i  1 − 1i −1 − 1i  1 + 1i 0 0 1 0 0 1 0 1 −1 − 1i  1 − 1i −1 + 1i −1 +1i 0 0 1 0 0 1 1 1 −1 − 1i  1 − 1i −1 + 1i  1 + 1i 0 0 1 0 1 0 0 0 −1 −1i  1 − 1i  1 − 1i −1 − 1i 0 0 1 0 1 0 1 0 −1 − 1i  1 − 1i  1 − 1i  1 −1i 0 0 1 0 1 1 0 0 −1 − 1i  1 − 1i  1 + 1i −1 − 1i 0 0 1 0 1 1 1 0 −1 −1i  1 − 1i  1 + 1i  1 − 1i 0 0 1 1 0 0 0 0 −1 − 1i  1 + 1i −1 − 1i −1 −1i 0 0 1 1 0 0 1 0 −1 − 1i  1 + 1i −1 − 1i  1 − 1i 0 0 1 1 0 1 0 1 −1 −1i  1 + 1i −1 + 1i −1 + 1i 0 0 1 1 0 1 1 1 −1 − 1i  1 + 1i −1 + 1i  1 +1i 0 0 1 1 1 0 0 0 −1 − 1i  1 + 1i  1 − 1i −1 − 1i 0 0 1 1 1 0 1 0 −1 −1i  1 + 1i  1 − 1i  1 − 1i 0 0 1 1 1 1 0 1 −1 − 1i  1 + 1i  1 + 1i −1 +1i 0 0 1 1 1 1 1 1 −1 − 1i  1 + 1i  1 + 1i  1 + 1i 0 1 0 0 0 0 0 1 −1 +1i − 1 − 1i −1 − 1i −1 + 1i 0 1 0 0 0 0 1 1 −1 + 1i − 1 − 1i −1 − 1i 1 + 1i 0 1 0 0 0 1 0 1 −1 + 1i − 1 − 1i −1 + 1i −1 + 1i 0 1 0 0 0 1 1 1−1 + 1i − 1 − 1i −1 + 1i  1 + 1i 0 1 0 0 1 0 0 0 −1 + 1i − 1 − 1i  1 −1i −1 − 1i 0 1 0 0 1 0 1 0 −1 + 1i − 1 − 1i  1 − 1i  1 − 1i 0 1 0 0 1 10 0 −1 + 1i − 1 − 1i  1 + 1i −1 − 1i 0 1 0 0 1 1 1 0 −1 + 1i − 1 − 1i 1 + 1i  1 − 1i 0 1 0 1 0 0 0 1 −1 + 1i − 1 + 1i −1 − 1i −1 + 1i 0 1 0 10 0 1 1 −1 + 1i − 1 + 1i −1 − 1i  1 + 1i 0 1 0 1 0 1 0 0 −1 + 1i − 1 +1i −1 + 1i −1 − 1i 0 1 0 1 0 1 1 0 −1 + 1i − 1 + 1i −1 + 1i  1 − 1i 0 10 1 1 0 0 1 −1 + 1i − 1 + 1i  1 − 1i −1 + 1i 0 1 0 1 1 0 1 1 −1 + 1i −1 + 1i  1 − 1i  1 + 1i 0 1 0 1 1 1 0 0 −1 + 1i − 1 + 1i  1 + 1i −1 − 1i0 1 0 1 1 1 1 0 −1 + 1i − 1 + 1i  1 + 1i  1 − 1i 0 1 1 0 0 0 0 0 −1 + 1i 1 − 1i −1 − 1i −1 − 1i 0 1 1 0 0 0 1 0 −1 + 1i  1 − 1i −1 − 1i  1 − 1i0 1 1 0 0 1 0 1 −1 + 1i  1 − 1i −1 + 1i −1 + 1i 0 1 1 0 0 1 1 1 −1 + 1i 1 − 1i −1 + 1i  1 + 1i 0 1 1 0 1 0 0 0 −1 + 1i  1 − 1i  1 − 1i −1 − 1i0 1 1 0 1 0 1 0 −1 + 1i  1 − 1i  1 − 1i  1 − 1i 0 1 1 0 1 1 0 1 −1 + 1i 1 − 1i  1 + 1i −1 + 1i 0 1 1 0 1 1 1 1 −1 + 1i  1 − 1i  1 + 1i  1 + 1i0 1 1 1 0 0 0 0 −1 + 1i  1 + 1i −1 − 1i −1 − 1i 0 1 1 1 0 0 1 0 −1 + 1i 1 + 1i −1 − 1i  1 − 1i 0 1 1 1 0 1 0 0 −1 + 1i  1 + 1i −1 + 1i −1 − 1i0 1 1 1 0 1 1 0 −1 + 1i  1 + 1i −1 + 1i  1 − 1i 0 1 1 1 1 0 0 1 −1 + 1i 1 + 1i  1 − 1i −1 + 1i 0 1 1 1 1 0 1 1 −1 + 1i  1 + 1i  1 − 1i  1 + 1i0 1 1 1 1 1 0 1 −1 + 1i  1 + 1i  1 + 1i −1 + 1i 0 1 1 1 1 1 1 1 −1 + 1i 1 + 1i  1 + 1i  1 + 1i 1 0 0 0 0 0 0 0  1 − 1i −1 − 1i −1 − 1i −1 − 1i1 0 0 0 0 0 1 0  1 − 1i −1 − 1i −1 − 1i  1 − 1i 1 0 0 0 0 1 0 0  1 − 1i−1 − 1i −1 + 1i −1 − 1i 1 0 0 0 0 1 1 0  1 − 1i −1 − 1i −1 + 1i  1 − 1i1 0 0 0 1 0 0 1  1 − 1i −1 − 1i  1 − 1i −1 + 1i 1 0 0 0 1 0 1 1  1 − 1i−1 − 1i  1 − 1i  1 + 1i 1 0 0 0 1 1 0 1  1 − 1i −1 − 1i  1 + 1i −1 + 1i1 0 0 0 1 1 1 1  1 − 1i −1 − 1i  1 + 1i  1 + 1i 1 0 0 1 0 0 0 0  1 − 1i−1 + 1i −1 − 1i −1 − 1i 1 0 0 1 0 0 1 0  1 − 1i −1 + 1i −1 − 1i  1 − 1i1 0 0 1 0 1 0 1  1 − 1i −1 + 1i −1 + 1i −1 + 1i 1 0 0 1 0 1 1 1  1 − 1i−1 + 1i −1 + 1i  1 + 1i 1 0 0 1 1 0 0 0  1 − 1i −1 + 1i  1 − 1i −1 − 1i1 0 0 1 1 0 1 0  1 − 1i −1 + 1i  1 − 1i  1 − 1i 1 0 0 1 1 1 0 1  1 − 1i−1 + 1i  1 + 1i −1 + 1i 1 0 0 1 1 1 1 1  1 − 1i −1 + 1i  1 + 1i  1 + 1i1 0 1 0 0 0 0 1  1 − 1i  1 − 1i −1 − 1i −1 + 1i 1 0 1 0 0 0 1 1  1 − 1i 1 − 1i −1 − 1i  1 + 1i 1 0 1 0 0 1 0 0  1 − 1i  1 − 1i −1 + 1i −1 − 1i1 0 1 0 0 1 1 0  1 − 1i  1 − 1i −1 + 1i  1 − 1i 1 0 1 0 1 0 0 1  1 − 1i 1 − 1i  1 − 1i −1 + 1i 1 0 1 0 1 0 1 1  1 − 1i  1 − 1i  1 − 1i  1 + 1i1 0 1 0 1 1 0 0  1 − 1i  1 − 1i  1 + 1i −1 − 1i 1 0 1 0 1 1 1 0  1 − 1i 1 − 1i  1 + 1i  1 − 1i 1 0 1 1 0 0 0 1  1 − 1i  1 + 1i −1 − 1i −1 + 1i1 0 1 1 0 0 1 1  1 − 1i  1 + 1i −1 − 1i  1 + 1i 1 0 1 1 0 1 0 1  1 − 1i 1 + 1i −1 + 1i −1 + 1i 1 0 1 1 0 1 1 1  1 − 1i  1 + 1i −1 + 1i  1 + 1i1 0 1 1 1 0 0 0  1 − 1i  1 + 1i  1 − 1i −1 − 1i 1 0 1 1 1 0 1 0  1 − 1i 1 + 1i  1 − 1i  1 − 1i 1 0 1 1 1 1 0 0  1 − 1i  1 + 1i  1 + 1i −1 − 1i1 0 1 1 1 1 1 0  1 − 1i  1 + 1i  1 + 1i  1 − 1i 1 1 0 0 0 0 0 0  1 + 1i−1 − 1i −1 − 1i −1 − 1i 1 1 0 0 0 0 1 0  1 + 1i −1 − 1i −1 − 1i  1 − 1i1 1 0 0 0 1 0 1  1 + 1i −1 − 1i −1 + 1i −1 + 1i 1 1 0 0 0 1 1 1  1 + 1i−1 − 1i −1 + 1i  1 + 1i 1 1 0 0 1 0 0 0  1 + 1i −1 − 1i  1 − 1i −1 − 1i1 1 0 0 1 0 1 0  1 + 1i −1 − 1i  1 − 1i  1 − 1i 1 1 0 0 1 1 0 1  1 + 1i−1 − 1i  1 + 1i −1 + 1i 1 1 0 0 1 1 1 1  1 + 1i −1 − 1i  1 + 1i  1 + 1i1 1 0 1 0 0 0 1  1 + 1i −1 + 1i −1 − 1i −1 + 1i 1 1 0 1 0 0 1 1  1 + 1i−1 + 1i −1 − 1i  1 + 1i 1 1 0 1 0 1 0 1  1 + 1i −1 + 1i −1 + 1i −1 + 1i1 1 0 1 0 1 1 1  1 + 1i −1 + 1i −1 + 1i  1 + 1i 1 1 0 1 1 0 0 0  1 + 1i−1 + 1i  1 − 1i −1 − 1i 1 1 0 1 1 0 1 0  1 + 1i −1 + 1i  1 − 1i  1 − 1i1 1 0 1 1 1 0 0  1 + 1i −1 + 1i  1 + 1i −1 − 1i 1 1 0 1 1 1 1 0  1 + 1i−1 + 1i  1 + 1i  1 − 1i 1 1 1 0 0 0 0 0  1 + 1i  1 − 1i −1 − 1i −1 − 1i1 1 1 0 0 0 1 0  1 + 1i  1 − 1i −1 − 1i  1 − 1i 1 1 1 0 0 1 0 0  1 + 1i 1 − 1i −1 + 1i −1 − 1i 1 1 1 0 0 1 1 0  1 + 1i  1 − 1i −1 + 1i  1 − 1i1 1 1 0 1 0 0 1  1 + 1i  1 − 1i  1 − 1i −1 + 1i 1 1 1 0 1 0 1 1  1 + 1i 1 − 1i  1 − 1i  1 + 1i 1 1 1 0 1 1 0 1  1 + 1i  1 − 1i  1 + 1i −1 + 1i1 1 1 0 1 1 1 1  1 + 1i  1 − 1i  1 + 1i  1 + 1i 1 1 1 1 0 0 0 1  1 + 1i 1 + 1i −1 − 1i −1 + 1i 1 1 1 1 0 0 1 1  1 + 1i  1 + 1i −1 − 1i  1 + 1i1 1 1 1 0 1 0 0  1 + 1i  1 + 1i −1 + 1i −1 − 1i 1 1 1 1 0 1 1 0  1 + 1i 1 + 1i −1 + 1i  1 − 1i 1 1 1 1 1 0 0 1  1 + 1i  1 + 1i  1 − 1i −1 + 1i1 1 1 1 1 0 1 1  1 + 1i  1 + 1i  1 − 1i  1 + 1i 1 1 1 1 1 1 0 0  1 + 1i 1 + 1i  1 + 1i −1 − 1i 1 1 1 1 1 1 1 0  1 + 1i  1 + 1i  1 + 1i  1 − 1i

These symbols have 4 possible states of polarization with the conditionthat the state of polarization on T₂ is either opposite to or differentfrom the state of polarization on T₁. The constellation has a highsymmetry. The structure is such that the each constellation point hasthe same number of neighbors. The neighbors are located at 8 differentEuclidean distances, as shown on the following table.

Euclidean Distance Number of neighboring symbols 2 4 2.82 12 3.46 28 438 4.47 28 4.89 12 5.29 4 5.65 1

Basically, every point of the constellation has 8 symbols at Euclideandistance of 2, 12 symbols at an Euclidean distance of 2.82, and so on.As for the modulation format of the previous embodiment, this structureis highly symmetrical, which yields good linear channel performance.

In the third exemplary embodiment, the modulation format is defined in8D: I, Q, polarization and two consecutive time slots. To map bits intosymbols 201-204, the following approach is used: from 4 information bitsof the data signal 101, referred to as [b₁, b₂, b₃, b₄], four overheadbits [b′₁, b′₂, b′₃, b′₄] are obtained using the following equation:b1′=b1XORb2XORb3b2′=b1XORb2XORb5b3′=b1XORb3XOR b5b4′=b2XORb3XOR b5

Thus, the set of [b₁, b₂, b₃, b′₁, b₄, b′₂, b′₃, b′₄] is obtained. Themapping is then done as follows. The first two bits [b₁ b₂] are used toselect a symbol 201-204 from the set 200 (QPSK constellation) shown inFIG. 2 . These symbols represents the 2 dimensions I and Q of the Xpolarization on the time slot T₁. Using the same approach, I and Qsymbols of Y polarization on T₁, X polarization on T₂ and finally Ypolarization on T₂ are obtained using the bits [b₃ b′₁], [b₄ b′₂] andfinally [b′₃ b′₄], respectively. At the end, a spectral efficiency of 2bits/transmission time slot (4 bits in 8 dimensions) is reached, and allobtained symbols 201-204 are given in the following table. The labelling(mapping from bits to symbols 201-204) of each symbol 201-204 is givenin the table as well, because as mentioned before, different labellingmay result on different linear channel performance (except forequivalent constellations related by symmetry).

Labelling (from the left to the right: 3 information bits, 1 parity bit,1 information bit, Time slot T₁ Time slot T₂ 3 parity bits) Xpolarization Y polarization X polarization Y polarization 00000011 −1 −1i −1 − 1i −1 − 1i +1 + 1i 00001100 −1 − 1i −1 − 1i +1 + 1i −1 − 1i00110000 −1 − 1i +1 + 1i −1 − 1i −1 − 1i 00111111 −1 − 1i +1 + 1i +1 +1i +1 + 1i 01010110 −1 + 1i −1 + 1i −1 + 1i +1 − 1i 01011001 −1 + 1i−1 + 1i +1 − 1i −1 + 1i 01100101 −1 + 1i +1 − 1i −1 + 1i −1 + 1i01101010 −1 + 1i +1 − 1i +1 − 1i +1 − 1i 10010101 +1 − 1i −1 + 1i −1 +1i −1 + 1i 10011010 +1 − 1i −1 + 1i +1 − 1i +1 − 1i 10100110 +1 − 1i +1− 1i −1 + 1i +1 − 1i 10101001 +1 − 1i +1 − 1i +1 − 1i −1 + 1i 11000000+1 + 1i −1 − 1i −1 − 1i −1 − 1i 11001111 +1 + 1i −1 − 1i +1 + 1i +1 + 1i11110011 +1 + 1i +1 + 1i −1 − 1i +1 + 1i 11111100 +1 + 1i +1 + 1i +1 +1i −1 − 1i

These symbols have 4 possible states of polarization with the conditionthat the state of polarization on T₂ is opposite to the state ofpolarization on T₁. The constellation has a high symmetry. The structureis such that the each constellation point has the same number ofneighbors. The neighbors are located at 2 different Euclidean distances,as shown on the following table.

Euclidean Distance Number of neighboring symbols 4 14 5.65 1

Basically, every point of the constellation has 14 symbols at Euclideandistance of 4 and one symbol at an Euclidean distance of 5.65. As forthe modulation format of the previous embodiment, this structure ishighly symmetrical, which yields good linear channel performance.

In the fourth exemplary embodiment, the modulation format is defined in8D: I, Q, polarization and two consecutive time slots. To map bits intosymbols 201-204, the following approach is used: from 6 information bitsof the data signal 101, referred to as [b₁, b₂, b₃, b₄, b₅, b₆], twooverhead bits [b′₁, b′₂] are obtained using the following equation:b1′=b2XORb3XORb5XOR(b1XORb2)AND(b3XORb4XORb5XORb6)XOR(b3XORb4)AND(b5XORb6)b2′=b1XORb4XORb6XOR(b1XORb2)AND(b3XORb4XORb5XORb6)XOR(b3XORb4)AND(b5XORb6)

Thus, the set of [b₁, b₂, b₃, b₄, b₅, b₆, b′₁, b′₂] is obtained. Themapping is then done as follows. The first two bits [b₁ b₂] are used toselect a symbol 201-204 from the set 200 (QPSK constellation) shown inFIG. 2 . These symbols represents the 2 dimensions I and Q of the Xpolarization on the time slot T₁. Using the same approach, I and Qsymbols of Y polarization on T₁, X polarization on T₂ and finally Ypolarization on T₂ are obtained using the bits [b₃ b₄], [b₅ b₆] andfinally [b′₁ b′₂], respectively. At the end, a spectral efficiency of 3bits/transmission time slot (6 bits in 8 dimensions) is reached, and allobtained symbols 201-204 are given in the following table. The labelling(mapping from bits to symbols 201-204) of each symbol 201-204 is givenin the table as well, because as mentioned before, different labellingmay result on different linear channel performance (except forequivalent constellations related by symmetry).

Labelling (from the left to the right: 6 information Time slot T₁ Timeslot T₂ bits and 2 parity bit) X polarization Y polarization Xpolarization Y polarization 00000011 −1 − 1i −1 − 1i −1 − 1i +1 + 1i00000110 −1 − 1i −1 − 1i −1 + 1i +1 − 1i 00001001 −1 − 1i −1 − 1i +1 −1i −1 + 1i 00001100 −1 − 1i −1 − 1i +1 + 1i −1 − 1i 00010010 −1 − 1i−1 + 1i −1 − 1i +1 − 1i 00010100 −1 − 1i −1 + 1i −1 + 1i −1 − 1i00011011 −1 − 1i −1 + 1i +1 − 1i +1 + 1i 00011101 −1 − 1i −1 + 1i +1 +1i −1 + 1i 00100001 −1 − 1i +1 − 1i −1 − 1i −1 + 1i 00100111 −1 − 1i +1− 1i −1 + 1i +1 + 1i 00101000 −1 − 1i +1 − 1i +1 − 1i −1 − 1i 00101110−1 − 1i +1 − 1i +1 + 1i +1 − 1i 00110000 −1 − 1i +1 + 1i −1 − 1i −1 − 1i00110101 −1 − 1i +1 + 1i −1 + 1i −1 + 1i 00111010 −1 − 1i +1 + 1i +1 −1i +1 − 1i 00111111 −1 − 1i +1 + 1i +1 + 1i +1 + 1i 01000001 −1 + 1i −1− 1i −1 − 1i −1 + 1i 01000111 −1 + 1i −1 − 1i −1 + 1i +1 + 1i 01001000−1 + 1i −1 − 1i +1 − 1i −1 − 1i 01001110 −1 + 1i −1 − 1i +1 + 1i +1 − 1i01010011 −1 + 1i −1 + 1i −1 − 1i +1 + 1i 01010110 −1 + 1i −1 + 1i −1 +1i +1 − 1i 01011001 −1 + 1i −1 + 1i +1 − 1i −1 + 1i 01011100 −1 + 1i−1 + 1i +1 + 1i −1 − 1i 01100000 −1 + 1i +1 − 1i −1 − 1i −1 − 1i01100101 −1 + 1i +1 − 1i −1 + 1i −1 + 1i 01101010 −1 + 1i +1 − 1i +1 −1i +1 − 1i 01101111 −1 + 1i +1 − 1i +1 + 1i +1 + 1i 01110010 −1 + 1i+1 + 1i −1 − 1i +1 − 1i 01110100 −1 + 1i +1 + 1i −1 + 1i −1 − 1i01111011 −1 + 1i +1 + 1i +1 − 1i +1 + 1i 01111101 −1 + 1i +1 + 1i +1 +1i −1 + 1i 10000010 +1 − 1i −1 − 1i −1 − 1i +1 − 1i 10000100 +1 − 1i −1− 1i −1 + 1i −1 − 1i 10001011 +1 − 1i −1 − 1i +1 − 1i +1 + 1i 10001101+1 − 1i −1 − 1i +1 + 1i −1 + 1i 10010000 +1 − 1i −1 + 1i −1 − 1i −1 − 1i10010101 +1 − 1i −1 + 1i −1 + 1i −1 + 1i 10011010 +1 − 1i −1 + 1i +1 −1i +1 − 1i 10011111 +1 − 1i −1 + 1i +1 + 1i +1 + 1i 10100011 +1 − 1i +1− 1i −1 − 1i +1 + 1i 10100110 +1 − 1i +1 − 1i −1 + 1i +1 − 1i 10101001+1 − 1i +1 − 1i +1 − 1i −1 + 1i 10101100 +1 − 1i +1 − 1i +1 + 1i −1 − 1i10110001 +1 − 1i +1 + 1i −1 − 1i −1 + 1i 10110111 +1 − 1i +1 + 1i −1 +1i +1 + 1i 10111000 +1 − 1i +1 + 1i +1 − 1i −1 − 1i 10111110 +1 − 1i+1 + 1i +1 + 1i +1 − 1i 11000000 +1 + 1i −1 − 1i −1 − 1i −1 − 1i11000101 +1 + 1i −1 − 1i −1 + 1i −1 + 1i 11001010 +1 + 1i −1 − 1i +1 −1i +1 − 1i 11001111 +1 + 1i −1 − 1i +1 + 1i +1 + 1i 11010001 +1 + 1i−1 + 1i −1 − 1i −1 + 1i 11010111 +1 + 1i −1 + 1i −1 + 1i +1 + 1i11011000 +1 + 1i −1 + 1i +1 − 1i −1 − 1i 11011110 +1 + 1i −1 + 1i +1 +1i +1 − 1i 11100010 +1 + 1i +1 − 1i −1 − 1i +1 − 1i 11100100 +1 + 1i +1− 1i −1 + 1i −1 − 1i 11101011 +1 + 1i +1 − 1i +1 − 1i +1 + 1i 11101101+1 + 1i +1 − 1i +1 + 1i −1 + 1i 11110011 +1 + 1i +1 + 1i −1 − 1i +1 + 1i11110110 +1 + 1i +1 + 1i −1 + 1i +1 − 1i 11111001 +1 + 1i +1 + 1i +1 −1i −1 + 1i 11111100 +1 + 1i +1 + 1i +1 + 1i −1 − 1i

These symbols have 4 possible states of polarization with the conditionthat the state of polarization on T₂ is opposite to the state ofpolarization on T₁. The constellation has a high symmetry. The structureis such that the each constellation point has the same number ofneighbors. The neighbors are located at 4 different Euclidean distances,as shown on the following table.

Euclidean Distance Number of neighboring symbols 2.82 12 4 38 4.89 125.65 1

Basically, every point of the constellation has 12 symbols at Euclideandistance of 2.82, 38 symbols at an Euclidean distance of 4, and so on.As for the modulation format of the previous embodiment, this structureis highly symmetrical, which yields good linear channel performance.

The four exemplary modulation formats presented above, provide theoptical transmitter 100 with a linear and nonlinear channel performancethat exceeds the state of the art. In particular, the exemplarymodulation format with a spectral efficiency of 2.5 bit/transmissiontime slot has a better nonlinear performance (found to be 0.35 dB higherin Q2 factor) than the one of a corresponding conventional solution,even though the linear performance is the same. The modulation formatwith spectral efficiency of 3.5 bit/transmission time slot has betterlinear and nonlinear performance. This is because the set 200 of symbols201-0204 (base constellation) has a higher Euclidian distance, whichgives better linear performance. As mentioned above, this can beachieved because the polarization-balance criterion is relaxed topolarization alternating. This allows using the base constellation asfor example shown in FIG. 2 . Modulation formats of spectralefficiencies of 2 and 3 bit/transmission time slot have the sameperformance of the state-of-the-art but are generated using Booleanequations, which simplify the mapper and demapper and allow lowcomplexity implementation, which results on low power consumingsolution.

FIG. 5 shows a method 500 of optically transmitting a data signal 101.The method 500 may be performed by the optical transmitter 100. Themethod comprises a step 501 of encoding 501 the data signal 101 byselecting based on a bit sequence a first symbol and a second symbolfrom a set 200 of four symbols 201-204 for each one of at least twotransmission time slots. This step 501 may be performed by the encoder102. The method 500 further comprises a step 502 of using in eachtransmission time slot the first symbol to modulate a first carrier waveand the second symbol to modulate a second carrier wave, andtransmitting the two carrier waves over orthogonal polarizations of anoptical carrier 104. This step 501 may be performed by the modulator103. Symbols 201-204 in consecutive transmission time slots havenon-identical polarization states.

The present invention has been described in conjunction with variousembodiments as examples as well as implementations. However, othervariations can be understood and effected by those persons skilled inthe art and practicing the claimed invention, from the studies of thedrawings, this disclosure and the independent claims. In the claims aswell as in the description the word “comprising” does not exclude otherelements or steps and the indefinite article “a” or “an” does notexclude a plurality. A single element or other unit may fulfill thefunctions of several entities or items recited in the claims. The merefact that certain measures are recited in the mutual different dependentclaims does not indicate that a combination of these measures cannot beused in an advantageous implementation.

What is claimed is:
 1. An optical transmitter for transmitting a data signal, the optical transmitter comprising: an encoder configured to encode the data signal by selecting a first symbol and a second symbol from a set of four symbols for each one of at least two transmission time slots; and a modulator configured to use in each transmission time slot the first symbol to modulate a first carrier wave and the second symbol to modulate a second carrier wave, and to transmit the first carrier wave and the second carrier wave over orthogonal polarizations of an optical carrier, wherein symbols in consecutive transmission time slots have non-identical polarization states, wherein the symbols correspond to the carrier waves and the transmission time slots, and the symbols are mapped to a bit sequence comprising the data signal and an overhead sequence, and Boolean equations are used to generate the overhead sequence from the data signal, wherein the optical transmitter is configured to transmit the data signal with a spectral efficiency of 3.5 bits per transmission time slot, wherein symbols in at least a subset of the consecutive transmission time slots have orthogonal polarization states, wherein the data signal has seven bits b1 . . . b7, and the encoder is configured to generate the bit sequence having eight bits b1 . . . b7, b′, wherein the overhead bit b′ is generated according to: b1′= b1XORb4XORb6XOR (b1XORb2)AND(b3XORb4XORb5XORb6)XOR (b3XORb4)AND(b5XORb6), and wherein for two of the consecutive transmission time slots T1 and T2, for two of the orthogonal polarizations X and Y of the optical carrier, and for a set of four QPSK symbols denoted −1−1i, −1+1i, 1−1i and 1+1i, the encoder is configured to select the symbols based on the data signal according to the following labelling: Labelling (from left to right) 7 bits b1 . . . b7, Time slot T₁ Time slot T₂ and 1 overhead X polar- Y polar- X polari- Y polar- bit b′ ization ization zation ization 0 0 0 0 0 0 0 1 −1 − 1i −1 − 1i −1 − 1i −1 + 1i 0 0 0 0 0 0 1 1 −1 − 1i −1 − 1i −1 − 1i  1 + 1i 0 0 0 0 0 1 0 0 −1 − 1i −1 − 1i −1 + 1i −1 − 1i 0 0 0 0 0 1 1 0 −1 − 1i −1 − 1i −1 + 1i  1 − 1i 0 0 0 0 1 0 0 1 −1 − 1i −1 − 1i  1 − 1i −1 + 1i 0 0 0 0 1 0 1 1 −1 − 1i −1 − 1i  1 − 1i  1 + 1i 0 0 0 0 1 1 0 0 −1 − 1i −1 − 1i  1 + 1i −1 − 1i 0 0 0 0 1 1 1 0 −1 − 1i −1 − 1i  1 + 1i  1 − 1i 0 0 0 1 0 0 0 0 −1 − 1i −1 + 1i −1 − 1i −1 − 1i 0 0 0 1 0 0 1 0 −1 − 1i −1 + 1i −1 − 1i  1 − 1i 0 0 0 1 0 1 0 0 −1 − 1i −1 + 1i −1 + 1i −1 − 1i 0 0 0 1 0 1 1 0 −1 − 1i −1 + 1i −1 + 1i  1 − 1i 0 0 0 1 1 0 0 1 −1 − 1i −1 + 1i  1 − 1i −1 + 1i 0 0 0 1 1 0 1 1 −1 − 1i −1 + 1i  1 − 1i  1 + 1i 0 0 0 1 1 1 0 1 −1 − 1i −1 + 1i  1 + 1i −1 + 1i 0 0 0 1 1 1 1 1 −1 − 1i −1 + 1i  1 + 1i  1 + 1i 0 0 1 0 0 0 0 1 −1 − 1i  1 − 1i −1 − 1i −1 + 1i 0 0 1 0 0 0 1 1 −1 − 1i  1 − 1i −1 − 1i  1 + 1i 0 0 1 0 0 1 0 1 −1 − 1i  1 − 1i −1 + 1i −1 + 1i 0 0 1 0 0 1 1 1 −1 − 1i  1 − 1i −1 + 1i  1 + 1i 0 0 1 0 1 0 0 0 −1 − 1i  1 − 1i  1 − 1i −1 − 1i 0 0 1 0 1 0 1 0 −1 − 1i  1 − 1i  1 − 1i  1 − 1i 0 0 1 0 1 1 0 0 −1 − 1i  1 − 1i  1 + 1i −1 − 1i 0 0 1 0 1 1 1 0 −1 − 1i  1 − 1i  1 + 1i  1 − 1i 0 0 1 1 0 0 0 0 −1 − 1i  1 + 1i −1 − 1i −1 − 1i 0 0 1 1 0 0 1 0 −1 − 1i  1 + 1i −1 − 1i  1 − 1i 0 0 1 1 0 1 0 1 −1 − 1i  1 + 1i −1 + 1i −1 + 1i 0 0 1 1 0 1 1 1 −1 − 1i  1 + 1i −1 + 1i  1 + 1i 0 0 1 1 1 0 0 0 −1 − 1i  1 + 1i  1 − 1i −1 − 1i 0 0 1 1 1 0 1 0 −1 − 1i  1 + 1i  1 − 1i  1 − 1i 0 0 1 1 1 1 0 1 −1 − 1i  1 + 1i  1 + 1i −1 + 1i 0 0 1 1 1 1 1 1 −1 − 1i  1 + 1i  1 + 1i  1 + 1i 0 1 0 0 0 0 0 1 −1 + 1i −1 − 1i −1 − 1i −1 + 1i 0 1 0 0 0 0 1 1 −1 + 1i −1 − 1i −1 − 1i  1 + 1i 0 1 0 0 0 1 0 1 −1 + 1i −1 − 1i −1 + 1i −1 + 1i 0 1 0 0 0 1 1 1 −1 + 1i −1 − 1i −1 + 1i  1 + 1i 0 1 0 0 1 0 0 0 −1 + 1i −1 − 1i  1 − 1i −1 − 1i 0 1 0 0 1 0 1 0 −1 + 1i −1 − 1i  1 − 1i  1 − 1i 0 1 0 0 1 1 0 0 −1 + 1i −1 − 1i  1 + 1i −1 − 1i 0 1 0 0 1 1 1 0 −1 + 1i −1 − 1i  1 + 1i  1 − 1i 0 1 0 1 0 0 0 1 −1 + 1i −1 + 1i −1 − 1i −1 + 1i 0 1 0 1 0 0 1 1 −1 + 1i −1 + 1i −1 − 1i  1 + 1i 0 1 0 1 0 1 0 0 −1 + 1i −1 + 1i −1 + 1i −1 − 1i 0 1 0 1 0 1 1 0 −1 + 1i −1 + 1i −1 + 1i  1 − 1i 0 1 0 1 1 0 0 1 −1 + 1i −1 + 1i  1 − 1i −1 + 1i 0 1 0 1 1 0 1 1 −1 + 1i −1 + 1i  1 − 1i  1 + 1i 0 1 0 1 1 1 0 0 −1 + 1i −1 + 1i  1 + 1i −1 − 1i 0 1 0 1 1 1 1 0 −1 + 1i −1 + 1i  1 + 1i  1 − 1i 0 1 1 0 0 0 0 0 −1 + 1i  1 − 1i −1 − 1i −1 − 1i 0 1 1 0 0 0 1 0 −1 + 1i  1 − 1i −1 − 1i  1 − 1i 0 1 1 0 0 1 0 1 −1 + 1i  1 − 1i −1 + 1i −1 + 1i 0 1 1 0 0 1 1 1 −1 + 1i  1 − 1i −1 + 1i  1 + 1i 0 1 1 0 1 0 0 0 −1 + 1i  1 − 1i  1 − 1i −1 − 1i 0 1 1 0 1 0 1 0 −1 + 1i  1 − 1i  1 − 1i  1 − 1i 0 1 1 0 1 1 0 1 −1 + 1i  1 − 1i  1 + 1i −1 + 1i 0 1 1 0 1 1 1 1 −1 + 1i  1 − 1i  1 + 1i  1 + 1i 0 1 1 1 0 0 0 0 −1 + 1i  1 + 1i −1 − 1i −1 − 1i 0 1 1 1 0 0 1 0 −1 + 1i  1 + 1i −1 − 1i  1 − 1i 0 1 1 1 0 1 0 0 −1 + 1i  1 + 1i −1 + 1i −1 − 1i 0 1 1 1 0 1 1 0 −1 + 1i  1 + 1i −1 + 1i  1 − 1i 0 1 1 1 1 0 0 1 −1 + 1i  1 + 1i  1 − 1i −1 + 1i 0 1 1 1 1 0 1 1 −1 + 1i  1 + 1i  1 − 1i  1 + 1i 0 1 1 1 1 1 0 1 −1 + 1i  1 + 1i  1 + 1i −1 + 1i 0 1 1 1 1 1 1 1 −1 + 1i  1 + 1i  1 + 1i  1 + 1i 1 0 0 0 0 0 0 0  1 − 1i −1 − 1i −1 − 1i −1 − 1i 1 0 0 0 0 0 1 0  1 − 1i −1 − 1i −1 − 1i  1 − 1i 1 0 0 0 0 1 0 0  1 − 1i −1 − 1i −1 + 1i −1 − 1i 1 0 0 0 0 1 1 0  1 − 1i −1 − 1i −1 + 1i  1 − 1i 1 0 0 0 1 0 0 1  1 − 1i −1 − 1i  1 − 1i −1 + 1i 1 0 0 0 1 0 1 1  1 − 1i −1 − 1i  1 − 1i  1 + 1i 1 0 0 0 1 1 0 1  1 − 1i −1 − 1i  1 + 1i −1 + 1i 1 0 0 0 1 1 1 1  1 − 1i −1 − 1i  1 + 1i  1 + 1i 1 0 0 1 0 0 0 0  1 − 1i −1 + 1i −1 − 1i −1 − 1i 1 0 0 1 0 0 1 0  1 − 1i −1 + 1i −1 − 1i  1 − 1i 1 0 0 1 0 1 0 1  1 − 1i −1 + 1i −1 + 1i −1 + 1i 1 0 0 1 0 1 1 1  1 − 1i −1 + 1i −1 + 1i  1 + 1i 1 0 0 1 1 0 0 0  1 − 1i −1 + 1i  1 − 1i −1 − 1i 1 0 0 1 1 0 1 0  1 − 1i −1 + 1i  1 − 1i  1 − 1i 1 0 0 1 1 1 0 1  1 − 1i −1 + 1i  1 + 1i −1 + 1i 1 0 0 1 1 1 1 1  1 − 1i −1 + 1i  1 + 1i  1 + 1i 1 0 1 0 0 0 0 1  1 − 1i  1 − 1i −1 − 1i −1 + 1i 1 0 1 0 0 0 1 1  1 − 1i  1 − 1i −1 − 1i  1 + 1i 1 0 1 0 0 1 0 0  1 − 1i  1 − 1i −1 + 1i −1 − 1i 1 0 1 0 0 1 1 0  1 − 1i  1 − 1i −1 + 1i  1 − 1i 1 0 1 0 1 0 0 1  1 − 1i  1 − 1i  1 − 1i −1 + 1i 1 0 1 0 1 0 1 1  1 − 1i  1 − 1i  1 − 1i  1 + 1i 1 0 1 0 1 1 0 0  1 − 1i  1 − 1i  1 + 1i −1 − 1i 1 0 1 0 1 1 1 0  1 − 1i  1 − 1i  1 + 1i  1 − 1i 1 0 1 1 0 0 0 1  1 − 1i  1 + 1i −1 − 1i −1 + 1i 1 0 1 1 0 0 1 1  1 − 1i  1 + 1i −1 − 1i  1 + 1i 1 0 1 1 0 1 0 1  1 − 1i  1 + 1i −1 + 1i −1 + 1i 1 0 1 1 0 1 1 1  1 − 1i  1 + 1i −1 + 1i  1 + 1i 1 0 1 1 1 0 0 0  1 − 1i  1 + 1i  1 − 1i −1 − 1i 1 0 1 1 1 0 1 0  1 − 1i  1 + 1i  1 − 1i  1 − 1i 1 0 1 1 1 1 0 0  1 − 1i  1 + 1i  1 + 1i −1 − 1i 1 0 1 1 1 1 1 0  1 − 1i  1 + 1i  1 + 1i  1 − 1i 1 1 0 0 0 0 0 0  1 + 1i −1 − 1i −1 − 1i −1 − 1i 1 1 0 0 0 0 1 0  1 + 1i −1 − 1i −1 − 1i  1 − 1i 1 1 0 0 0 1 0 1  1 + 1i −1 − 1i −1 + 1i −1 + 1i 1 1 0 0 0 1 1 1  1 + 1i −1 − 1i −1 + 1i  1 + 1i 1 1 0 0 1 0 0 0  1 + 1i −1 − 1i  1 − 1i −1 − 1i 1 1 0 0 1 0 1 0  1 + 1i −1 − 1i  1 − 1i  1 − 1i 1 1 0 0 1 1 0 1  1 + 1i −1 − 1i  1 + 1i −1 + 1i 1 1 0 0 1 1 1 1  1 + 1i −1 − 1i  1 + 1i  1 + 1i 1 1 0 1 0 0 0 1  1 + 1i −1 + 1i −1 − 1i −1 + 1i 1 1 0 1 0 0 1 1  1 + 1i −1 + 1i −1 − 1i  1 + 1i 1 1 0 1 0 1 0 1  1 + 1i −1 + 1i −1 + 1i −1 + 1i 1 1 0 1 0 1 1 1  1 + 1i −1 + 1i −1 + 1i  1 + 1i 1 1 0 1 1 0 0 0  1 + 1i −1 + 1i  1 − 1i −1 − 1i 1 1 0 1 1 0 1 0  1 + 1i −1 + 1i  1 − 1i  1 − 1i 1 1 0 1 1 1 0 0  1 + 1i −1 + 1i  1 + 1i −1 − 1i 1 1 0 1 1 1 1 0  1 + 1i −1 + 1i  1 + 1i  1 − 1i 1 1 1 0 0 0 0 0  1 + 1i  1 − 1i −1 − 1i −1 − 1i 1 1 1 0 0 0 1 0  1 + 1i  1 − 1i −1 − 1i  1 − 1i 1 1 1 0 0 1 0 0  1 + 1i  1 − 1i −1 + 1i −1 − 1i 1 1 1 0 0 1 1 0  1 + 1i  1 − 1i −1 + 1i  1 − 1i 1 1 1 0 1 0 0 1  1 + 1i  1 − 1i  1 − 1i −1 + 1i 1 1 1 0 1 0 1 1  1 + 1i  1 − 1i  1 − 1i  1 + 1i 1 1 1 0 1 1 0 1  1 + 1i  1 − 1i  1 + 1i −1 + 1i 1 1 1 0 1 1 1 1  1 + 1i  1 − 1i  1 + 1i  1 + 1i 1 1 1 1 0 0 0 1  1 + 1i  1 + 1i −1 − 1i −1 + 1i 1 1 1 1 0 0 1 1  1 + 1i  1 + 1i −1 − 1i  1 + 1i 1 1 1 1 0 1 0 0  1 + 1i  1 + 1i −1 + 1i −1 − 1i 1 1 1 1 0 1 1 0  1 + 1i  1 + 1i −1 + 1i  1 − 1i 1 1 1 1 1 0 0 1  1 + 1i  1 + 1i  1 − 1i −1 + 1i 1 1 1 1 1 0 1 1  1 + 1i  1 + 1i  1 − 1i  1 + 1i 1 1 1 1 1 1 0 0  1 + 1i  1 + 1i  1 + 1i −1 − 1i 1 1 1 1 1 1 1 0  1 + 1i  1 + 1i  1 + 1i   1 − 1i.


2. The optical transmitter according to claim 1, wherein the encoder is configured to select the symbols from a quadrature phase shift keying (QPSK) base constellation.
 3. The optical transmitter according to claim 1, wherein the modulator is configured to modulate an in-phase component and a quadrature component of each carrier wave.
 4. The optical transmitter according to claim 1, wherein the symbols in at least a subset of the consecutive transmission time slots have anti-parallel polaritzation states.
 5. The optical transmitter according to claim 1, wherein: the encoder is configured to generate the bit sequence based on the data signal, and the data signal comprises less bits than the bit sequence.
 6. The optical transmitter according to claim 5, wherein the encoder is configured to perform at least one Boolean operation based on at least two bits of the data signal to obtain at least one overhead bit of the overhead sequence, and to generate the bit sequence based on the bits of the data signal and at least one overhead bit.
 7. An optical transmission system, the optical transmission comprising: the optical transmitter according to claim 1; and an optical receiver configured to receive the data signal, wherein the optical receiver is configured to receive and decode the modulated carrier waves of the optical carrier to obtain the data signal.
 8. A method of optically transmitting a data signal, the method comprising: encoding the data signal by selecting a first symbol and a second symbol from a set of four symbols for each one of at least two transmission time slots; using, in each of the transmission time slots, the first symbol to modulate a first carrier wave and the second symbol to modulate a second carrier wave; and transmitting the two carrier waves over orthogonal polarizations of an optical carrier, wherein symbols in consecutive ones of the transmission time slots have non-identical polarization states, wherein the symbols are corresponding to the carrier waves and the transmission time slots, and the symbols are mapped to a bit sequence comprising the data signal and an overhead sequence, and wherein Boolean equations are used to generate the overhead sequence from the data signal, and wherein the method further comprises transmitting the data signal with a spectral efficiency of 3.5 bits per transmission time slot, wherein symbols in at least a subset of the consecutive transmission time slots have orthogonal polarization states, wherein the data signal has seven bits b1 . . . b7, and the method further comprises generating the bit sequence having eight bits b1 . . . b7, b′, wherein the overhead bit b′ is generated according to: b1′= b1XORb4XORb6XOR (b1XORb2)AND(b3XORb4XORb5XORb6)XOR (b3XORb4)AND(b5XORb6), and wherein for two of the consecutive transmission time slots T1 and T2, for two of the orthogonal polarizations X and Y of the optical carrier, and for a set of four QPSK symbols denoted −1−1i, −1+1i, 1−1i and 1+1i, the method comprises selecting the symbols based on the data signal according to the following labelling: Labelling (from left to right) 7 bits b1 . . . b7, Time slot T₁ Time slot T₂ and 1 overhead bit b′ X polarization Y polarization X polarization Y polarization 0 0 0 0 0 0 0 1 −1 − 1i −1 − 1i −1 − 1i −1 + 1i 0 0 0 0 0 0 1 1 −1 − 1i −1 − 1i −1 − 1i  1 + 1i 0 0 0 0 0 1 0 0 −1 − 1i −1 − 1i −1 + 1i −1 − 1i 0 0 0 0 0 1 1 0 −1 − 1i −1 − 1i −1 + 1i  1 − 1i 0 0 0 0 1 0 0 1 −1 − 1i −1 − 1i  1 − 1i −1 + 1i 0 0 0 0 1 0 1 1 −1 − 1i −1 − 1i  1 − 1i  1 + 1i 0 0 0 0 1 1 0 0 −1 − 1i −1 − 1i  1 + 1i −1 − 1i 0 0 0 0 1 1 1 0 −1 − 1i −1 − 1i  1 + 1i  1 − 1i 0 0 0 1 0 0 0 0 −1 − 1i −1 + 1i −1 − 1i −1 − 1i 0 0 0 1 0 0 1 0 −1 − 1i −1 + 1i −1 − 1i  1 − 1i 0 0 0 1 0 1 0 0 −1 − 1i −1 + 1i −1 + 1i −1 − 1i 0 0 0 1 0 1 1 0 −1 − 1i −1 + 1i −1 + 1i  1 − 1i 0 0 0 1 1 0 0 1 −1 − 1i −1 + 1i  1 − 1i −1 + 1i 0 0 0 1 1 0 1 1 −1 − 1i −1 + 1i  1 − 1i  1 + 1i 0 0 0 1 1 1 0 1 −1 − 1i −1 + 1i  1 + 1i −1 + 1i 0 0 0 1 1 1 1 1 −1 − 1i −1 + 1i  1 + 1i  1 + 1i 0 0 1 0 0 0 0 1 −1 − 1i  1 − 1i −1 − 1i −1 + 1i 0 0 1 0 0 0 1 1 −1 − 1i  1 − 1i −1 − 1i  1 + 1i 0 0 1 0 0 1 0 1 −1 − 1i  1 − 1i −1 + 1i −1 + 1i 0 0 1 0 0 1 1 1 −1 − 1i  1 − 1i −1 + 1i  1 + 1i 0 0 1 0 1 0 0 0 −1 − 1i  1 − 1i  1 − 1i −1 − 1i 0 0 1 0 1 0 1 0 −1 − 1i  1 − 1i  1 − 1i  1 − 1i 0 0 1 0 1 1 0 0 −1 − 1i  1 − 1i  1 + 1i −1 − 1i 0 0 1 0 1 1 1 0 −1 − 1i  1 − 1i  1 + 1i  1 − 1i 0 0 1 1 0 0 0 0 −1 − 1i  1 + 1i −1 − 1i −1 − 1i 0 0 1 1 0 0 1 0 −1 − 1i  1 + 1i −1 − 1i  1 − 1i 0 0 1 1 0 1 0 1 −1 − 1i  1 + 1i −1 + 1i −1 + 1i 0 0 1 1 0 1 1 1 −1 − 1i  1 + 1i −1 + 1i  1 + 1i 0 0 1 1 1 0 0 0 −1 − 1i  1 + 1i  1 − 1i −1 − 1i 0 0 1 1 1 0 1 0 −1 − 1i  1 + 1i  1 − 1i  1 − 1i 0 0 1 1 1 1 0 1 −1 − 1i  1 + 1i  1 + 1i −1 + 1i 0 0 1 1 1 1 1 1 −1 − 1i  1 + 1i  1 + 1i  1 + 1i 0 1 0 0 0 0 0 1 −1 + 1i −1 − 1i −1 − 1i −1 + 1i 0 1 0 0 0 0 1 1 −1 + 1i −1 − 1i −1 − 1i  1 + 1i 0 1 0 0 0 1 0 1 −1 + 1i −1 − 1i −1 + 1i −1 + 1i 0 1 0 0 0 1 1 1 −1 + 1i −1 − 1i −1 + 1i  1 + 1i 0 1 0 0 1 0 0 0 −1 + 1i −1 − 1i  1 − 1i −1 − 1i 0 1 0 0 1 0 1 0 −1 + 1i −1 − 1i  1 − 1i  1 − 1i 0 1 0 0 1 1 0 0 −1 + 1i −1 − 1i  1 + 1i −1 − 1i 0 1 0 0 1 1 1 0 −1 + 1i −1 − 1i  1 + 1i  1 − 1i 0 1 0 1 0 0 0 1 −1 + 1i −1 + 1i −1 − 1i −1 + 1i 0 1 0 1 0 0 1 1 −1 + 1i −1 + 1i −1 − 1i  1 + 1i 0 1 0 1 0 1 0 0 −1 + 1i −1 + 1i −1 + 1i −1 − 1i 0 1 0 1 0 1 1 0 −1 + 1i −1 + 1i −1 + 1i  1 − 1i 0 1 0 1 1 0 0 1 −1 + 1i −1 + 1i  1 − 1i −1 + 1i 0 1 0 1 1 0 1 1 −1 + 1i −1 + 1i  1 − 1i  1 + 1i 0 1 0 1 1 1 0 0 −1 + 1i −1 + 1i  1 + 1i −1 − 1i 0 1 0 1 1 1 1 0 −1 + 1i −1 + 1i  1 + 1i  1 − 1i 0 1 1 0 0 0 0 0 −1 + 1i  1 − 1i −1 − 1i −1 − 1i 0 1 1 0 0 0 1 0 −1 + 1i  1 − 1i −1 − 1i  1 − 1i 0 1 1 0 0 1 0 1 −1 + 1i  1 − 1i −1 + 1i −1 + 1i 0 1 1 0 0 1 1 1 −1 + 1i  1 − 1i −1 + 1i  1 + 1i 0 1 1 0 1 0 0 0 −1 + 1i  1 − 1i  1 − 1i −1 − 1i 0 1 1 0 1 0 1 0 −1 + 1i  1 − 1i  1 − 1i  1 − 1i 0 1 1 0 1 1 0 1 −1 + 1i  1 − 1i  1 + 1i −1 + 1i 0 1 1 0 1 1 1 1 −1 + 1i  1 − 1i  1 + 1i  1 + 1i 0 1 1 1 0 0 0 0 −1 + 1i  1 + 1i −1 − 1i −1 − 1i 0 1 1 1 0 0 1 0 −1 + 1i  1 + 1i −1 − 1i  1 − 1i 0 1 1 1 0 1 0 0 −1 + 1i  1 + 1i −1 + 1i −1 − 1i 0 1 1 1 0 1 1 0 −1 + 1i  1 + 1i −1 + 1i  1 − 1i 0 1 1 1 1 0 0 1 −1 + 1i  1 + 1i  1 − 1i −1 + 1i 0 1 1 1 1 0 1 1 −1 + 1i  1 + 1i  1 − 1i  1 + 1i 0 1 1 1 1 1 0 1 −1 + 1i  1 + 1i  1 + 1i −1 + 1i 0 1 1 1 1 1 1 1 −1 + 1i  1 + 1i  1 + 1i  1 + 1i 1 0 0 0 0 0 0 0  1 − 1i −1 − 1i −1 − 1i −1 − 1i 1 0 0 0 0 0 1 0  1 − 1i −1 − 1i −1 − 1i  1 − 1i 1 0 0 0 0 1 0 0  1 − 1i −1 − 1i −1 + 1i −1 − 1i 1 0 0 0 0 1 1 0  1 − 1i −1 − 1i −1 + 1i  1 − 1i 1 0 0 0 1 0 0 1  1 − 1i −1 − 1i  1 − 1i −1 + 1i 1 0 0 0 1 0 1 1  1 − 1i −1 − 1i  1 − 1i  1 + 1i 1 0 0 0 1 1 0 1  1 − 1i −1 − 1i  1 + 1i −1 + 1i 1 0 0 0 1 1 1 1  1 − 1i −1 − 1i  1 + 1i  1 + 1i 1 0 0 1 0 0 0 0  1 − 1i −1 + 1i −1 − 1i −1 − 1i 1 0 0 1 0 0 1 0  1 − 1i −1 + 1i −1 − 1i  1 − 1i 1 0 0 1 0 1 0 1  1 − 1i −1 + 1i −1 + 1i −1 + 1i 1 0 0 1 0 1 1 1  1 − 1i −1 + 1i −1 + 1i  1 + 1i 1 0 0 1 1 0 0 0  1 − 1i −1 + 1i  1 − 1i −1 − 1i 1 0 0 1 1 0 1 0  1 − 1i −1 + 1i  1 − 1i  1 − 1i 1 0 0 1 1 1 0 1  1 − 1i −1 + 1i  1 + 1i −1 + 1i 1 0 0 1 1 1 1 1  1 − 1i −1 + 1i  1 + 1i  1 + 1i 1 0 1 0 0 0 0 1  1 − 1i  1 − 1i −1 − 1i −1 + 1i 1 0 1 0 0 0 1 1  1 − 1i  1 − 1i −1 − 1i  1 + 1i 1 0 1 0 0 1 0 0  1 − 1i  1 − 1i −1 + 1i −1 − 1i 1 0 1 0 0 1 1 0  1 − 1i  1 − 1i −1 + 1i  1 − 1i 1 0 1 0 1 0 0 1  1 − 1i  1 − 1i  1 − 1i −1 + 1i 1 0 1 0 1 0 1 1  1 − 1i  1 − 1i  1 − 1i  1 + 1i 1 0 1 0 1 1 0 0  1 − 1i  1 − 1i  1 + 1i −1 − 1i 1 0 1 0 1 1 1 0  1 − 1i  1 − 1i  1 + 1i  1 − 1i 1 0 1 1 0 0 0 1  1 − 1i  1 + 1i −1 − 1i −1 + 1i 1 0 1 1 0 0 1 1  1 − 1i  1 + 1i −1 − 1i  1 + 1i 1 0 1 1 0 1 0 1  1 − 1i  1 + 1i −1 + 1i −1 + 1i 1 0 1 1 0 1 1 1  1 − 1i  1 + 1i −1 + 1i  1 + 1i 1 0 1 1 1 0 0 0  1 − 1i  1 + 1i  1 − 1i −1 − 1i 1 0 1 1 1 0 1 0  1 − 1i  1 + 1i  1 − 1i  1 − 1i 1 0 1 1 1 1 0 0  1 − 1i  1 + 1i  1 + 1i −1 − 1i 1 0 1 1 1 1 1 0  1 − 1i  1 + 1i  1 + 1i  1 − 1i 1 1 0 0 0 0 0 0  1 + 1i −1 − 1i −1 − 1i −1 − 1i 1 1 0 0 0 0 1 0  1 + 1i −1 − 1i −1 − 1i  1 − 1i 1 1 0 0 0 1 0 1  1 + 1i −1 − 1i −1 + 1i −1 + 1i 1 1 0 0 0 1 1 1  1 + 1i −1 − 1i −1 + 1i  1 + 1i 1 1 0 0 1 0 0 0  1 + 1i −1 − 1i  1 − 1i −1 − 1i 1 1 0 0 1 0 1 0  1 + 1i −1 − 1i  1 − 1i  1 − 1i 1 1 0 0 1 1 0 1  1 + 1i −1 − 1i  1 + 1i −1 + 1i 1 1 0 0 1 1 1 1  1 + 1i −1 − 1i  1 + 1i  1 + 1i 1 1 0 1 0 0 0 1  1 + 1i −1 + 1i −1 − 1i −1 + 1i 1 1 0 1 0 0 1 1  1 + 1i −1 + 1i −1 − 1i  1 + 1i 1 1 0 1 0 1 0 1  1 + 1i −1 + 1i −1 + 1i −1 + 1i 1 1 0 1 0 1 1 1  1 + 1i −1 + 1i −1 + 1i  1 + 1i 1 1 0 1 1 0 0 0  1 + 1i −1 + 1i  1 − 1i −1 − 1i 1 1 0 1 1 0 1 0  1 + 1i −1 + 1i  1 − 1i  1 − 1i 1 1 0 1 1 1 0 0  1 + 1i −1 + 1i  1 + 1i −1 − 1i 1 1 0 1 1 1 1 0  1 + 1i −1 + 1i  1 + 1i  1 − 1i 1 1 1 0 0 0 0 0  1 + 1i  1 − 1i −1 − 1i −1 − 1i 1 1 1 0 0 0 1 0  1 + 1i  1 − 1i −1 − 1i  1 − 1i 1 1 1 0 0 1 0 0  1 + 1i  1 − 1i −1 + 1i −1 − 1i 1 1 1 0 0 1 1 0  1 + 1i  1 − 1i −1 + 1i  1 − 1i 1 1 1 0 1 0 0 1  1 + 1i  1 − 1i  1 − 1i −1 + 1i 1 1 1 0 1 0 1 1  1 + 1i  1 − 1i  1 − 1i  1 + 1i 1 1 1 0 1 1 0 1  1 + 1i  1 − 1i  1 + 1i −1 + 1i 1 1 1 0 1 1 1 1  1 + 1i  1 − 1i  1 + 1i  1 + 1i 1 1 1 1 0 0 0 1  1 + 1i  1 + 1i −1 − 1i −1 + 1i 1 1 1 1 0 0 1 1  1 + 1i  1 + 1i −1 − 1i  1 + 1i 1 1 1 1 0 1 0 0  1 + 1i  1 + 1i −1 + 1i −1 − 1i 1 1 1 1 0 1 1 0  1 + 1i  1 + 1i −1 + 1i  1 − 1i 1 1 1 1 1 0 0 1  1 + 1i  1 + 1i  1 − 1i −1 + 1i 1 1 1 1 1 0 1 1  1 + 1i  1 + 1i  1 − 1i  1 + 1i 1 1 1 1 1 1 0 0  1 + 1i  1 + 1i  1 + 1i −1 − 1i 1 1 1 1 1 1 1 0  1 + 1i  1 + 1i  1 + 1i   1 − 1i. 